On the growth of mth derivatives of algebraic polynomials in the weighted Lebesgue space

ABSTRACT

In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found.

KEYWORDS:

• Algebraic polynomial
• quasiconformal mapping
• smooth curve
• quasismooth curve
• asymptotic conformal curve

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 30C30
• 30E10
• 30C70

1. Introduction

Let $\mathbb{C}$ be a complex plane and $\overline{\mathbb{C}}:=\mathbb{C}\cup \left\{\mathrm{\infty }\right\};G\subset \mathbb{C}$ be a bounded Jordan region with boundary $L:=\mathrm{\partial }G$ (without loss of generality, let $0\in G)$; $\mathrm{\Omega }:=\overline{\mathbb{C}}\mathrm{\setminus }\overline{G}=extL$. For $t\in \mathbb{C}$ and $\delta >0$, let $\mathrm{\Delta }(t,\delta ):=\{w\in \mathbb{C}:|w-t|>\delta \};\mathrm{\Delta }:=\mathrm{\Delta }(0,1)$. Let $\mathrm{\Phi }:\mathrm{\Omega }\to \mathrm{\Delta }$ be the univalent conformal mapping normalized by $\mathrm{\Phi }\left(\mathrm{\infty }\right)=\mathrm{\infty }$ and $\underset{z\to \mathrm{\infty }}{lim}\frac{\mathrm{\Phi }\left(z\right)}{z}>0$; $\mathrm{\Psi }:={\mathrm{\Phi }}^{-1}$. For$R>1$, we take $L_R:=\{z:|\mathrm{\Phi }\left(z\right)|=R\}$, $G_R:=int{L}_R$, and ${\mathrm{\Omega }}_R:=ext{L}_R$.

Let ${\mathrm{\wp }}_n$ denotes the class of all algebraic polynomials $P_n\left(z\right)$ of degree at most $n\in \mathbb{N}$.

Let $\{z_j{\}}_{j=1}^l\in L$ be the fixed system of distinct points. For some fixed $R_0$, $1<R_0<\mathrm{\infty }$, and $z\in {\overline{G}}_{R_0}$, consider generalized Jacobi weight function $h\left(z\right)$: (1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) where ${\gamma }_j>-1$, for all$j=1,2,\dots ,l,z\in G_{R_0}$ and some constant $c_0\left(L\right)>0$, $h_0\left(z\right)\ge c_0\left(L\right)>0$.

For each $0<p\le \mathrm{\infty }$ and rectifiable Jordan curve $L=\mathrm{\partial }G$, we introduce: (2) $\begin{array}{rl}{\Vert P_n\Vert }_p& :={\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)}:={\left({\int }_{L}h\right(z\left){\left|P_n\right(z\left)\right|}^p\left|dz\right|\right)}^{1/p}<\mathrm{\infty },0<p<\mathrm{\infty },\\ {\Vert P_n\Vert }_{\mathrm{\infty }}& :={\Vert P_n\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}(1,L)}:=\underset{z\in L}{max}\left|P_n\right(z\left)\right|,p=\mathrm{\infty };{\mathcal{L}}_p(1,L)=:{\mathcal{L}}_p\left(L\right).\end{array}$(2) In the theory of approximations of a function of a complex variable, the following Bernstein–Walsh inequality is often used [1]: (3) ${\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},\mathrm{\forall }{P}_n\in {\mathrm{\wp }}_n.$(3) It follows that the quantity $\Vert P_n{\Vert }_{\mathrm{\infty }}$ has the same order of growth in $\overline{G}$ and ${\overline{G}}_{1+c{n}^{-1}}$ with respect to n for all constants $c:=c\left(G\right)>0$.

An analogue of this inequality in space ${\mathcal{L}}_p(h,L)$ is the following inequality [2]: ${\Vert P_n\Vert }_{{\mathcal{L}}_p\left(L_R\right)}\le {\left|\mathrm{\Phi }\right(z\left)\right|}^{n+\frac{1}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p\left(L\right)},\mathrm{\forall }{P}_n\in {\mathrm{\wp }}_n,p>0.$This estimate has been generalized in [3, Lemma 2.4] for weight function $h\left(z\right)\ne 1$, defined as in (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ), as follows: (4) ${\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L_R)}\le R^{n+\frac{1+{\gamma }^{\ast }}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)},{\gamma }^{\ast }=max\{0;{\gamma }_j:1\le j\le l\}.$(4) If we replace the curve L to the region G and define two-dimensional analogues of the quantities (2(2) $\begin{array}{rl}{\Vert P_n\Vert }_p& :={\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)}:={\left({\int }_{L}h\right(z\left){\left|P_n\right(z\left)\right|}^p\left|dz\right|\right)}^{1/p}<\mathrm{\infty },0<p<\mathrm{\infty },\\ {\Vert P_n\Vert }_{\mathrm{\infty }}& :={\Vert P_n\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}(1,L)}:=\underset{z\in L}{max}\left|P_n\right(z\left)\right|,p=\mathrm{\infty };{\mathcal{L}}_p(1,L)=:{\mathcal{L}}_p\left(L\right).\end{array}$(2) ) (we denotes them by $\Vert P_n{\Vert }_{A_p(h,G)}$, $\Vert P_n{\Vert }_{A_p(1,G)}$ and $A_p\left(G\right)$ respectively), then for them we can also indicate the corresponding estimate of the type (4(4) ${\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L_R)}\le R^{n+\frac{1+{\gamma }^{\ast }}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)},{\gamma }^{\ast }=max\{0;{\gamma }_j:1\le j\le l\}.$(4) ). For this, first we will give the following definition.

For any $\delta >0$ and arbitrary $t,w\in \mathbb{C}$ let $B(t,\delta ):=\{t:|t-w|<\delta \}$ and $\phi :G\to B:=B(0,1)$ be a conformal and univalent map which is normalized by $\phi \left(0\right)=0$ and ${\phi }^{\mathrm{\prime }}\left(0\right)>0$; $\psi :={\phi }^{-1}$.

Definition 1.1

[4, p.286]

A bounded Jordan region G is called a κ-quasidisk, $0\le \kappa <1$, if any conformal mapping ψ can be extended to a K-quasiconformal, $K=\frac{1+\kappa }{1-\kappa }$, homeomorphism of the plane $\overline{\mathbb{C}}$ on $\overline{\mathbb{C}}$. In that case the curve $L:=\mathrm{\partial }G$ is called a κ-quasicircle. The region G (curve L) is called a quasidisk (quasicircle), if it is κ-quasidisk (κ-quasicircle) with some $0\le \kappa <1$.

Denote by $Q\left(\kappa \right),0\le \kappa <1$, class of κ-quasidisks and say that $L=\mathrm{\partial }G\in Q\left(\kappa \right)$, if $G\in Q\left(\kappa \right),0\le \kappa <1$. Further, we denote that $G\in Q(L\in Q)$, if $G\in Q\left(\kappa \right)(L\in Q(\kappa \left)\right)$ for some $0\le \kappa <1$. Quasicircles can be non-rectifiable (see, for example, [[5],[6, p.104]]). Since the object of study will be integrals along a curve, then we will say that $L\in \tilde{Q}\left(\kappa \right),0\le \kappa <1$, if $L\in Q\left(\kappa \right)$ and $L:=\mathrm{\partial }G$ is rectifiable. Correspondingly, $G\in \tilde{Q}$ $(L\in \tilde{Q})$, if $G\in \tilde{Q}\left(\kappa \right)(L\in \tilde{Q}(\kappa \left)\right)$ for some $0\le \kappa <1$.

The analogue of the estimates (3(3) ${\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},\mathrm{\forall }{P}_n\in {\mathrm{\wp }}_n.$(3) ) and (4(4) ${\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L_R)}\le R^{n+\frac{1+{\gamma }^{\ast }}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)},{\gamma }^{\ast }=max\{0;{\gamma }_j:1\le j\le l\}.$(4) ) for arbitrary quasidisks and $h\left(z\right)$ defined as in (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ) for the $\Vert P_n{\Vert }_{{}_{A_p(h,G)}}$ can be given as follows [7]: (5) ${\Vert P_n\Vert }_{{}_{A_p(h,G_R)}}\le c_1{R}^{{\ast }^{n+\frac{1}{p}}}{\Vert P_n\Vert }_{{}_{A_p(h,G)}},R>1,p>0,$(5) where $R^{\ast }:=1+c_2(R-1),c_2>0$ and $c_1:=c_1(G,p,c_2)>0$ constants, independent of n and R. This estimate was generalized for arbitrary Jordan region G and $P_n\in {\mathrm{\wp }}_n$ in [8, Theorem 1.1] as follows: ${\Vert P_n\Vert }_{{}_{A_p\left(G_R\right)}}\le c{R}^{n+\frac{2}{p}}{\Vert P_n\Vert }_{{}_{A_p\left(G_{R_1}\right)}},R>R_1=1+\frac{1}{n},p>0,$where $c=\left(\frac{2}{e^p-1}{)}^{\frac{1}{p}}\right[1+O\left(\frac{1}{n}\right)],n\to \mathrm{\infty }$, is asymptotically sharp constant.

N. Stylianopoulos [9] replaced the norm $\Vert P_n{\Vert }_{C\left(\overline{G}\right)}$ with norm $\Vert P_n{\Vert }_{A_2\left(G\right)}$ on the right-hand side of (3(3) ${\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},\mathrm{\forall }{P}_n\in {\mathrm{\wp }}_n.$(3) ) and found a new version of the Bernstein–Walsh Lemma: Let $L\in \tilde{Q}$. Then there exists a constant $C=C\left(L\right)>0$ depending only on L such that $\left|P_n\right(z\left)\right|\le C\frac{\sqrt{n}}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_{A_2\left(G\right)}{\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1},z\in \mathrm{\Omega },$holds for every $P_n\in {\mathrm{\wp }}_n$, where  $\mathrm{d}(z,L):=inf\left\{\right|\zeta -z|:\zeta \in L\}$.

In this paper, we continue the study of the problem on uniform and pointwise estimates of the derivatives $\left|P_n^{\left(m\right)}\right(z\left)\right|$, $m\ge 0$, in $\overline{G_R}$ and ${\mathrm{\Omega }}_R$, $R\ge 1$, and the following types of estimates will be found: (6) $\left|P_n^{\left(m\right)}\right(z\left)\right|\le c_4{\Vert P_n\Vert }_p\{\begin{array}{ll}{\lambda }_n(L,h,p,m)& z\in \overline{G_R},\\ {\eta }_n(L,h,p,m,z,m),& z\in {\mathrm{\Omega }}_R,\end{array}$(6) where ${\lambda }_n(.)$ and ${\eta }_n(.)\to \mathrm{\infty }$, as $n\to \mathrm{\infty }$, depending on the properties of the L, h.

Analogous results of (6(6) $\left|P_n^{\left(m\right)}\right(z\left)\right|\le c_4{\Vert P_n\Vert }_p\{\begin{array}{ll}{\lambda }_n(L,h,p,m)& z\in \overline{G_R},\\ {\eta }_n(L,h,p,m,z,m),& z\in {\mathrm{\Omega }}_R,\end{array}$(6) )-type for m = 0, different weight function h, in unbounded region were obtained in [3,9–22,23, p.418–428,24] and others.

Estimates of the (6(6) $\left|P_n^{\left(m\right)}\right(z\left)\right|\le c_4{\Vert P_n\Vert }_p\{\begin{array}{ll}{\lambda }_n(L,h,p,m)& z\in \overline{G_R},\\ {\eta }_n(L,h,p,m,z,m),& z\in {\mathrm{\Omega }}_R,\end{array}$(6) )-type for $z\in \overline{G}$, for the norms $\Vert P_n{\Vert }_{{\mathcal{L}}_p(h,L)}$ or $\Vert P_n{\Vert }_{A_p(h,G)}$, p>0, for some h ($h\left(z\right)\equiv 1$ or $h\left(z\right)\ne 1$) was studied since the beginning of the twentieth century [25–27] and has been studied in [7,23, p.418–428,24,28–36,37, Sect. 5.3,38,39, p.122–133,40] (see also the references cited therein) and others.

2. Definitions and main results

Throughout this paper, $c,c_0,c_1,c_2,\dots$ are positive and ${ϵ}_0,{ϵ}_1,{ϵ}_2,\dots$ are sufficiently small positive constants (generally, different in different relations), which depends on G in general and, on parameters inessential for the argument, otherwise, the dependence will be explicitly stated. For any $k\ge 0$ and m>k, notation $i=\overline{k,m}$ means $i=k,k+1,\dots ,m$.

Let S be rectifiable Jordan curve or arc and let $z=z\left(s\right),s\in [0,|S\left|\right],\left|S\right|:=mesS$, be the natural parametrization of S. A Jordan curve or arc is called smooth, if S has a continuous tangent $\theta \left(z\right):=\theta \left(z\right(s\left)\right)$ at every point $z\left(s\right)$. The class of such curves or arcs is denoted by $C_{\theta }$.

Let $z_1$, $z_2$ be arbitrary points on L and $l(z_1,z_2)$ denotes the subarc of L of shorter diameter with endpoints $z_1$ and $z_2$. The curve L is a quasicircle if and only if (three-point property). $L(z;z_{1,}{z}_2):=\underset{z_1,z_2\in L,z\in l(z_1,z_2)}{sup}\frac{|z_1-z|+|z-z_2|}{|z_1-z_2|}<\mathrm{\infty }$Lesley [41, p.341] said that the curve L is ‘c-quasiconformal’, if there exists the constant c>0, independent from points $z_1$, $z_2$ and z such that $L(z;z_{1,}{z}_2)\le c$. The Jordan curve L is called asymptotically conformal [42, 43], if $L(z;z_{1,}{z}_2)\to 1$, as $|z_1-z_2|\to 0$. According to the geometric criteria of quasiconformality of the curves [ [43, p.107],[44, p.81]], every asymptotically conformal curve is a quasicircle. Every smooth curve is asymptotically conformal but corners are not allowed. The asymptotically conformal curves can be non-rectifiable.

Following [4, p.163], we say that a bounded Jordan curve L is λ-quasismooth (in the sense of Lavrentiev) curve, if for every pair $z_1,z_2\in L$, there exists a constant $\lambda :=\lambda \left(L\right)\ge 1,$ such that $\left|l\right(z_1,z_2\left)\right|\le \lambda |z_1-z_2|,z_1,z_2\in L,$where $\left|l\right(z_1,z_2\left)\right|$ is the linear measure (length) of $l(z_1,z_2)$. The region G is called a λ-quasismooth region, if $L=\mathrm{\partial }G$ is a λ-quasismooth curve.

According to the ‘three-point’ criterion [6, p.100], every piecewise $C_{\theta }$-curve (without cusps) and quasismoth curve are quasiconformal.

In this work, we will try to get the result for more general curves, also including the above class of curves. For this we need to give the following definitions of quasidisks with some general functional conditions.

Definition 2.1

We say that $L=\mathrm{\partial }G\in Q_{\alpha }$, if $L$ is a quasicircle and $\mathrm{\Phi }\in H^{\alpha }\left(\overline{\mathrm{\Omega }}\right)$ for some $0<\alpha \le 1$.

Additionally, we say that $L\in {\tilde{Q}}_{\alpha },0<\alpha \le 1,$ if $L\in Q_{\alpha }$ and L is rectifiable.

We note, that the class $Q_{\alpha }$ is sufficiently large. A detailed account of it and the related topics are contained in [?, 41,45] (see also the references cited therein).

Definition 2.2

We say that $L=\mathrm{\partial }G\in Q_{\alpha }^{\text{β}}(L\in {\tilde{Q}}_{\alpha }^{\text{β}})$, if $L\in Q(L\in \tilde{Q})$, $\mathrm{\Phi }\in H^{\alpha }\left(\overline{\mathrm{\Omega }}\right)$ and $\mathrm{\Psi }\in H^{\text{β}}\left(\right|w|\ge 1)$ for some $0<\alpha \le 1$ and $0<\text{β}\le 1$.

We note, that the class $Q_{\alpha }$ and $Q_{\alpha }^{\text{β}}$ (also ${\tilde{Q}}_{\alpha }^{\text{β}}$) is sufficiently large. This can be seen from the following:

1. If L is a piecewise Dini-smooth curve and the largest exterior (interior) angle on L has opening $\vartheta \pi$ ($\text{β}\pi$), $0<\vartheta \le 1(1\le \nu <2)$, then $L\in {\tilde{Q}}_{\vartheta }(L\in {\tilde{Q}}_{\frac{1}{2-\nu }})$ [[41],[43, p.52]]. If L is a smooth curve having continuous tangent line, then $G\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for all $0<\alpha ,\text{β}<1$.

2. If G is ‘L-shaped’ region, then $G\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for $\alpha =\frac{2}{3}$ and $\text{β}=\frac{1}{2}$.

3. If L is quasismooth, then $G\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for $\alpha =\frac{1}{2}(1-\frac{1}{\pi }\arcsin \frac{1}{c}{)}^{-1}$ and $\text{β}=\frac{2}{(1+c{)}^2}$ [45,46].

4. If L is ‘c-quasiconformal’, then $G\in Q_{\alpha }^{\text{β}}$ for $\alpha =\frac{\pi }{2(\pi -\arcsin \frac{1}{c})}$ and $\text{β}=\frac{2(\arcsin \frac{1}{c}{)}^2}{\pi (\pi -\arcsin \frac{1}{c})}$ [41].

5. If L is an asymptotically conformal curve, then $G\in Q_{\alpha }^{\text{β}}$ for all $\alpha ,\text{β}<1$ [41].

Clearly, that $Q_{\alpha }^{\text{β}}\subset Q_{\alpha }({\tilde{Q}}_{\alpha }^{\text{β}}\subset {\tilde{Q}}_{\alpha })$ for any $0<\alpha \le 1$ and $0<\text{β}\le 1$. The class $Q_{\alpha }^{\text{β}}\left({\tilde{Q}}_{\alpha }^{\text{β}}\right)$ is defined as a subclass of the class $Q_{\alpha }\left({\tilde{Q}}_{\alpha }\right)$ with the additional functional condition. It turns out that more accurate estimates can be obtained for this class of curves.

For $0<{\delta }_j<{\delta }_0:=\frac{1}{4}min\left\{\right|z_i-z_j|:i,j=1,2,\dots ,l,i\ne j\}$, let $\mathrm{\Omega }(z_j,{\delta }_j):=\mathrm{\Omega }\cap \{z:|z-z_j|\le {\delta }_j\};\delta :=\underset{1\le j\le l}{min}{\delta }_j$; For $L=\mathrm{\partial }G$ we set: $U_{\mathrm{\infty }}(L,\delta ):=\bigcup _{\zeta \in L}U(\zeta ,\delta )$-infinite open cover of the curve $L;U_N(L,\delta ):=\bigcup _{j=1}^N{U}_j(L,\delta )\subset U_{\mathrm{\infty }}(L,\delta )$-finite open cover of the curve L; $\mathrm{\Omega }\left(\delta \right):=\mathrm{\Omega }(L,\delta ):=\mathrm{\Omega }\cap U_N(L,\delta ),\hat{\mathrm{\Omega }}:=\mathrm{\Omega }\mathrm{\setminus }\mathrm{\Omega }\left(\delta \right)$; ${\mathrm{\Omega }}_R\left(\delta \right):=\mathrm{\Omega }(L_R,\delta ):={\mathrm{\Omega }}_R\cap U_N(L_R,\delta )$, ${\hat{\mathrm{\Omega }}}_R:={\mathrm{\Omega }}_R\mathrm{\setminus }{\mathrm{\Omega }}_R\left(\delta \right)$.

Now, we start to formulate the new results. Firstly we give a recurrent estimate for $\left|P_n^{\left(m\right)}\right(z\left)\right|$, $m=1,2,\dots$.

Theorem 2.1

Let p>1; $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and every $m=1,2,\dots$, we have: (7) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_1\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left\{\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_{n,p}^1\right(z,m)+\sum _{j=1}^m{C}_m^j{B}_{n,j}^1(z\left)\left|P_n^{(m-j)}\left(z\right)\right|\right\},$(7) where $c_1=c_1(L,\gamma ,m,p)>0$ is a constant independent of n and z; $\begin{array}{rl}& A_{n,p}^1(z,m)\\ & :=\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\begin{array}{rl}{B}_{n,j}^1\left(z\right)& :=n^{\frac{j}{\alpha }},j=\overline{1,m},\\ & \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}\\ & A_{n,p}^1(z,m)\\ & :=\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}{B}_{n,j}^1\left(z\right):=n,j=\overline{1,m},\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}$and (8) ${\gamma }^{\ast }:=max\{0;{\gamma }_k,k=\overline{1,l}\}.$(8)

Theorem 2.2

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and every $m=1,2,\dots$, we have: (9) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_2\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left\{\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_{n,p}^1\right(z,m)+\sum _{j=1}^m{C}_m^j{B}_{n,j}^1(z\left)\left|P_n^{(m-j)}\left(z\right)\right|\right\},$(9) where $c_2=c_2(L,\gamma ,m,p)>0$ is a constant independent of n and z; $\begin{array}{rl}& A_{n,p}^1(z,m):=n^{(\frac{{\gamma }^{\ast }}{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},B_{n,j}^1\left(z\right):=n^{\frac{j+1}{\alpha }-\text{β}},j=1,2,\dots ,m,\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ & A_{n,p}^1(z,m)\\ & :=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{\alpha }},& \gamma >-1,\end{array}\begin{array}{rl}{B}_{n,j}^1\left(z\right)& :=n^{1-\text{β}},j=\overline{1,m},\\ & \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right);\end{array}\end{array}$and ${\gamma }^{\ast }$ is defined as in (8(8) ${\gamma }^{\ast }:=max\{0;{\gamma }_k,k=\overline{1,l}\}.$(8) ).

The estimates (7(7) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_1\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left\{\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_{n,p}^1\right(z,m)+\sum _{j=1}^m{C}_m^j{B}_{n,j}^1(z\left)\left|P_n^{(m-j)}\left(z\right)\right|\right\},$(7) ) and (9(9) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_2\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left\{\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_{n,p}^1\right(z,m)+\sum _{j=1}^m{C}_m^j{B}_{n,j}^1(z\left)\left|P_n^{(m-j)}\left(z\right)\right|\right\},$(9) ) allow us to obtain sequentially estimates for $\left|P_n^{\left(m\right)}\right(z\left)\right|$, $m\ge 2$. First, we will take m = 2 and j = 1, 2. Then, in these cases we need estimates for $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ and $\left|P_n\right(z\left)\right|$. Estimates for $m\ge 3$ are carried out sequentially by applying the estimates (7(7) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_1\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left\{\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_{n,p}^1\right(z,m)+\sum _{j=1}^m{C}_m^j{B}_{n,j}^1(z\left)\left|P_n^{(m-j)}\left(z\right)\right|\right\},$(7) ) and (9(9) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le c_2\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left\{\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_{n,p}^1\right(z,m)+\sum _{j=1}^m{C}_m^j{B}_{n,j}^1(z\left)\left|P_n^{(m-j)}\left(z\right)\right|\right\},$(9) ).

2.1. Estimate for $\left|P_n\right(z\left)\right|$

Theorem 2.3

Let p>1; $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and $z\in {\mathrm{\Omega }}_R$, we have: (10) $\left|P_n\left(z\right)\right|\le c_3\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^3,$(10) where $c_3=c_3(L,\gamma ,p)>0$ is a constant independent of n and z; $A_{n,p}^3:=\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& -1<\gamma \le 1+\alpha .\end{array}$

Theorem 2.4

[11, Th. 1.6 (Cor. 1.7)]

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and $z\in {\mathrm{\Omega }}_R$, we have: (11) $\left|P_n\left(z\right)\right|\le c_4\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^3,$(11) where $c_4=c_4(L,\gamma ,p)>0$ is a constant independent of n and z; $A_{n,p}^3:=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha .\end{array}$

2.2. Estimate for $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$

According to Theorems 2.1, 2.2 and 2.3, we can give an estimate for $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ at each point $z\in {\mathrm{\Omega }}_R$.

Theorem 2.5

Let p>1; $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: (12) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_5\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^4\left(z\right),$(12) where $c_5=c_5(L,\gamma ,p)>0$ is a constant independent of n and z; $\begin{array}{rl}{A}_{n,p}^4\left(z\right)& :=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma +1}{p\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{\frac{1}{\alpha }}{(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{1}{\alpha }},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }}& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{1}{\alpha }},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ A_{n,p}^4\left(z\right)& :=\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }+1},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n{(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{2-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{2-\frac{1}{p}},}& p>1,& -1<\gamma \le 1+\alpha ,\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$

Theorem 2.6

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: (13) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_6\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^5\left(z\right),$(13) where $c_6=c_6(L,\gamma ,p)>0$ is a constant independent of n and z; $\begin{array}{rl}& A_{n,p}^4\left(z\right)\\ & :=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{2}{\alpha }-(2-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha ,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ & A_{n,p}^4\left(z\right):=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+1-(2-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})}{(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha ,\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$

2.3. Estimate for $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\right(z\left)\right|$

Considering the estimates obtained for $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ (Theorems 2.5, 2.6) and for $\left|P_n\right(z\left)\right|$ (Theorem 2.3) in Theorems 2.1, 2.2 respectively, we have the following two theorems:

Theorem 2.7

Let p>1; $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: (14) $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|\le c_7\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^6\left(z\right),$(14) where $c_7=c_7(L,\gamma ,p)>0$ is a constant independent of n and z; $\begin{array}{rl}{A}_{n,p}^6\left(z\right)& :=\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{\frac{2}{\alpha }}{(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{2}{\alpha }},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{2}{\alpha }},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ A_{n,p}^6\left(z\right)& :=\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }+2},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^2{(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{3-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{3-\frac{1}{p}},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$

Theorem 2.8

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and $z\in {\mathrm{\Omega }}_R$, we have: (15) $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|\le c_8\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^7\left(z\right),$(15) where $c_8=c_8(L,\gamma ,p)>0$ is a constant independent of n and z; $\begin{array}{rl}& A_{n,p}^7\left(z\right):=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{4}{\alpha }-(3-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})}}\\ {(n\ln n)}^{1-\frac{1}{p}},& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& \begin{array}{l}p>1\\ \ge {\displaystyle \frac{\gamma +\alpha }{2+\alpha -2\alpha \text{β}}},\end{array}& \{\begin{array}{l}0<\gamma \le 2\\ (1-\alpha \text{β})\le \alpha ,\\ {\displaystyle \alpha \ge \frac{2}{1+2\text{β}};}\\ \text{β}\ge \frac{1}{2},\end{array}\\ {\displaystyle n^{(\frac{\gamma }{p}+2)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& \begin{array}{l}1<p\\ <{\displaystyle \frac{\gamma +\alpha }{2+\alpha -2\alpha \text{β}}},\end{array}& \{\begin{array}{l}2(1-\alpha \text{β})\\ <\gamma \le \alpha ,\\ {\displaystyle \alpha <\frac{2}{1+2\text{β}};}\\ \text{β}\ge \frac{1}{2},\end{array}\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& \begin{array}{l}p\ge {\displaystyle \frac{\gamma +\alpha }{2+\alpha -2\alpha \text{β}}}\\ >1,\end{array}& \{\begin{array}{l}2(1-\alpha \text{β})\\ <\gamma \le \alpha ,\\ {\displaystyle \alpha \ge \frac{2}{1+2\text{β}};}\\ \text{β}\ge \frac{1}{2},\end{array}\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ & A_{n,p}^7\left(z\right)\\ & :=\{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{2}{\alpha }-(3-\frac{1}{p})\text{β}+1},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \alpha <\gamma \le 2\\ & & -2\alpha \text{β}+2\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+1}}& & \\ {(n\ln n)}^{1-\frac{1}{p}},& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},& p\ge {\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}>1& 0<\gamma \le \alpha ;\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p\ge {\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}>1& \gamma >2-2\alpha \text{β}+2\alpha ,\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<{\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}& \gamma >2-2\alpha \text{β}+2\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p>1\ge {\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}& 0<\gamma \le \alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$

3. Estimate $\left|P_n^{\left(m\right)}\right(z\left)\right|,m\ge 1$, for bounded regions

Now, we can state estimates for $\left|P_n^{\left(m\right)}\right(z\left)\right|$, $m\ge 1$, in the bounded regions of the classes ${\tilde{Q}}_{\alpha }$ and ${\tilde{Q}}_{\alpha }^{\text{β}}$.

Theorem 3.1

Let p>1; $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and every $m=1,2,\dots$, we have: (16) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_9{n}^{(\frac{{\gamma }^{\ast }+1}{p}+m)\frac{1}{\alpha }}{\Vert P_n\Vert }_p,$(16) where $c_9=c_9(L,\gamma ,m,p)>0$ is a constant independent of n and z, and ${\gamma }^{\ast }$ is defined as in (8(8) ${\gamma }^{\ast }:=max\{0;{\gamma }_k,k=\overline{1,l}\}.$(8) ).

Theorem 3.2

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, and every $m=1,2,\dots$, we have: (17) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_{10}{n}^{(\frac{{\gamma }^{\ast }}{p}+m+1)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,$(17) where $c_{10}=c_{10}(L,\gamma ,m,p)>0$ is a constant independent of n and z, and ${\gamma }^{\ast }$ is defined as in (8(8) ${\gamma }^{\ast }:=max\{0;{\gamma }_k,k=\overline{1,l}\}.$(8) ).

Remark 3.1

The inequalities (16(16) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_9{n}^{(\frac{{\gamma }^{\ast }+1}{p}+m)\frac{1}{\alpha }}{\Vert P_n\Vert }_p,$(16) ) and (17(17) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_{10}{n}^{(\frac{{\gamma }^{\ast }}{p}+m+1)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,$(17) ) is sharp.

4. Estimates for $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ and $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\right(z\left)\right|$ in the whole complex plane

According to (3(3) ${\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},\mathrm{\forall }{P}_n\in {\mathrm{\wp }}_n.$(3) ) (applied to the polynomial $Q_{n-1}\left(z\right):=P_n^{\mathrm{\prime }}\left(z\right)$), the estimations (16(16) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_9{n}^{(\frac{{\gamma }^{\ast }+1}{p}+m)\frac{1}{\alpha }}{\Vert P_n\Vert }_p,$(16) ) and (17(17) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_{10}{n}^{(\frac{{\gamma }^{\ast }}{p}+m+1)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,$(17) ) are true also for the points $z\in {\overline{G}}_R$, $R=1+{ϵ}_0{n}^{-1}$, with a different constant. Therefore, combining these estimations with (12(12) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_5\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^4\left(z\right),$(12) ), (13(13) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_6\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^5\left(z\right),$(13) ), (14(14) $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|\le c_7\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^6\left(z\right),$(14) ), (15(15) $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|\le c_8\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^7\left(z\right),$(15) ), we will obtain an estimation on the growth for $|P_n^{\mathrm{\prime }}\left(z\right)|$ and $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\right(z\left)\right|$, respectively, in the whole complex plane:

Theorem 4.1

Let p>1; $G\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_{11}{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+1)\frac{1}{\alpha }},}& z\in {\overline{G}}_R,\\ {\displaystyle \frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{A}_{n,p}^4\left(z\right),}& z\in {\mathrm{\Omega }}_R,\end{array}$where $c_{11}=c_{11}(L,\gamma ,p)>0$ is a constant independent of n and z; $A_{n,p}^4\left(z\right)$ defined as in Theorem 2.5 for all $z\in {\mathrm{\Omega }}_R$.

Theorem 4.2

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_{12}{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+1)\frac{1}{\alpha }},}& z\in {\overline{G}}_R,\\ {\displaystyle \frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{A}_{n,p}^5\left(z\right),}& z\in {\mathrm{\Omega }}_R,\end{array}$where $c_{12}=c_{12}(L,\gamma ,p)>0$ is a constant independent of n and z; $A_{n,p}^5\left(z\right)$ defined as in Theorem 2.6 for all $z\in {\mathrm{\Omega }}_R$.

Theorem 4.3

Let p>1; $G\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|\le c_{13}{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+2)\frac{1}{\alpha }},}& z\in {\overline{G}}_R,\\ {\displaystyle \frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{A}_{n,p}^6\left(z\right),}& z\in {\mathrm{\Omega }}_R,\end{array}$where $c_{13}=c_{13}(L,\gamma ,p)>0$ is a constant independent of n and z; $A_{n,p}^6\left(z\right)$ is defined as in Theorem 2.7 for all $z\in {\mathrm{\Omega }}_R$.

Theorem 4.4

Let p>1; $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$ and $h\left(z\right)$ be defined by (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any $P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$, we have: $\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|\le c_{14}{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+2)\frac{1}{\alpha }},}& z\in {\overline{G}}_R,\\ {\displaystyle \frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{A}_{n,p}^7\left(z\right),}& z\in {\mathrm{\Omega }}_R,\end{array}$where $c_{14}=c_{14}(G,\gamma ,p)>0$ is a constant independent of n and z; $A_{n,p}^7\left(z\right)$ defined as in Theorem 2.8 for all $z\in {\mathrm{\Omega }}_R$.

Thus, using Theorems 2.1, 2.2 and estimating $\left|P_n^{\left(m\right)}\right(z\left)\right|$ sequentially for each $m\ge 3$, and combining the obtained estimates with Theorems 3.1 and 3.2, we can obtain estimates for the $\left|P_n^{\left(m\right)}\right(z\left)\right|$, for each points $z\in \mathbb{C}$.

5. Some auxiliary results

Throughout this paper we denote ‘$a⪯b$’ and ‘$a\approx b$’ are equivalent to $a\le cb$ and $c_{1}a\le b\le c_{2}a$ for some constants $c,c_1,c_2$, respectively.

Lemma 5.1

[47]

Let G be a quasidisk, $z_1\in L$, $z_2,z_3\in \mathrm{\Omega }\cap \{z:|z-z_1|⪯\mathrm{d}(z_1,L_{r_0}\left)\right\}$; $w_j=\mathrm{\Phi }\left(z_j\right)$, j = 1, 2, 3. Then

1. The statements $|z_1-z_2|⪯|z_1-z_3|$ and $|w_1-w_2|⪯|w_1-w_3|$ are equivalent. Therefore, $|z_1-z_2|\approx |z_1-z_3|$ and $|w_1-w_2|\approx |w_1-w_3|$ also are equivalent.

2. If $|z_1-z_2|⪯|z_1-z_3|$, then ${\left|\frac{w_1-w_3}{w_1-w_2}\right|}^{c_1}⪯\left|\frac{z_1-z_3}{z_1-z_2}\right|⪯{\left|\frac{w_1-w_3}{w_1-w_2}\right|}^{c_2},$where $0<r_0<1$ a constant, depending on G.

Corollary 5.1

Under the conditions of Lemma 5.1, we have: ${|w_1-w_2|}^{c_1}⪯|z_1-z_2|⪯{|w_1-w_2|}^{ϵ},$where $ϵ=ϵ\left(G\right)<1$.

Lemma 5.2

Let $L\in Q_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$. Then, for all $w_1,w_2:\left|w_1\right|\ge 1,\left|w_2\right|\ge 1$, we have: $\left|\mathrm{\Psi }\right(w_1)-\mathrm{\Psi }(w_2\left)\right|⪰{|w_1-w_2|}^{\frac{1}{\alpha }}.$

This fact follows from an appropriate result for the mapping $f\in \sum \left(\kappa \right)$ [4, p.287] and estimation for ${\mathrm{\Psi }}^{\mathrm{\prime }}$ [48, Th.2.8]: (18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18)

Lemma 5.3

[28]

Let L = G be a rectifiable Jordan curve and $P_n\left(z\right)$, $\mathrm{deg}{P}_n\le n,n=1,2,\dots$, be arbitrary polynomial and weight function $h\left(z\right)$ satisfy the condition (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then for any R>1, p>0 and $n=1,2,\dots$ ${\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L_R)}\le R^{n+\frac{1+{\gamma }^{\ast }}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)},{\gamma }^{\ast }=max\{0;{\gamma }_j:1\le j\le l\}.$

6. Proofs of theorems

Proofs

Proofs of Theorems 2.1 and 2.2

The proofs of Theorems 2.1 and 2.2 will be simultaneously.

Let $L\in {\tilde{Q}}_{\alpha }^{\text{β}}\subset {\tilde{Q}}_{\alpha }$, $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$, and let $R=1+\frac{1}{n}$, $R_1:=1+\frac{R-1}{2}$. For $z\in \mathrm{\Omega }$, let us define$H_n\left(z\right):=\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}$. Then, the mth derivative of $H_{n,p}\left(z\right)$ has a representation: $\begin{array}{rl}{H}_n^{\left(m\right)}\left(z\right)& =\sum _{j=0}^m{C}_m^j{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}\\ P_n^{(m-j)}\left(z\right)& =\frac{P_n^{\left(m\right)}\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}+\sum _{j=1}^m{C}_m^j{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}{P}_n^{(m-j)}\left(z\right),\end{array}$where $C_m^j:=\frac{m(m-1)\dots (m-j+1)}{j!}$. Passing to the modules in both parts, we get: (19) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\{\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|+\sum _{j=1}^m{C}_m^j\left|{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}\right|\left|P_n^{(m-j)}\left(z\right)\right|\}.$(19) Therefore, to estimate $\left|P_n^{\left(m\right)}\right(z\left)\right|$ for $z\in \mathrm{\Omega }$, it suffices to estimate the statements: (A) $\left|\right(P_n\left(z\right)\cdot {\mathrm{\Phi }}^{-n-1}\left(z\right){)}^{\left(m\right)}|$, $m=1,2,\dots$; (B) $\left|\right({\mathrm{\Phi }}^{-n-1}\left(z\right){)}^{\left(j\right)}|,$$j=\overline{1,m}$.

Now let us start with evaluations (A) and (B) for the cases $L\in {\tilde{Q}}_{\alpha }$ and $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$, separately.

(A) Since the function $H_n\left(z\right)$ is analytic in Ω, continuous on $\overline{\mathrm{\Omega }}$ and $H_n\left(\mathrm{\infty }\right)=0$, then Cauchy integral representation for the mth derivatives gives: $H_n^{\left(m\right)}\left(z\right)=-\frac{1}{2\pi i}{\int }_{L_{R_1}}{H}_n\left(\zeta \right)\frac{\mathrm{d}\zeta }{{(\zeta -z)}^{m+1}},z\in {\mathrm{\Omega }}_R,m\ge 1.$

Then, (20) $\begin{array}{rl}\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{m+1}}\\ & \le \frac{1}{2\pi \mathrm{d}(z,L_{R_1})}{\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m}.\end{array}$(20) Denote by (21) $A_n\left(z\right):={\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m},$(21) and estimate this integral. For this, we give some notations.

Let $w_j:=\mathrm{\Phi }\left(z_j\right),{\phi }_j:=\arg w_j$. Without loss of generality, we will assume that ${\phi }_l<2\pi$. For $\eta :=min\{{\eta }_j,j=\overline{1,l}\}$, where ${\eta }_j=\underset{t\in \mathrm{\partial }\mathrm{\Phi }\left(\mathrm{\Omega }\right(z_j,{\delta }_j\left)\right)}{min}|t-w_j|>0$, let us set: $\begin{array}{rl}{\mathrm{\Delta }}_j\left({\eta }_j\right)& :=\{t:|t-w_j|\le {\eta }_j\}\subset \mathrm{\Phi }\left(\mathrm{\Omega }\right(z_j,{\delta }_j\left)\right),\\ \mathrm{\Delta }\left(\eta \right)& :=\bigcup _{j=1}^l{\mathrm{\Delta }}_j\left(\eta \right),{\hat{\mathrm{\Delta }}}_j=\mathrm{\Delta }\mathrm{\setminus }\mathrm{\Delta }\left({\eta }_j\right);\hat{\mathrm{\Delta }}\left(\eta \right):=\mathrm{\Delta }\mathrm{\setminus }\mathrm{\Delta }\left(\eta \right);{\mathrm{\Delta }}_1^{\mathrm{\prime }}:={\mathrm{\Delta }}_1^{\mathrm{\prime }}\left(1\right),\\ {\mathrm{\Delta }}_1^{\mathrm{\prime }}\left(\rho \right)& :=\{t=R{e}^{i\theta }:R\ge \rho >1,\frac{{\phi }_0+{\phi }_1}{2}\le \theta <\frac{{\phi }_1+{\phi }_2}{2}\},\\ {\mathrm{\Delta }}_j^{\mathrm{\prime }}& :={\mathrm{\Delta }}_j^{\mathrm{\prime }}\left(1\right),{\mathrm{\Delta }}_j^{\mathrm{\prime }}\left(\rho \right):=\{t=R{e}^{i\theta }:R\ge \rho >1,\frac{{\phi }_{j-1}+{\phi }_j}{2}\le \theta <\frac{{\phi }_j+{\phi }_0}{2}\},\\ & j=2,3\dots ,l,\\ \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{\phi }_0& =2\pi -{\phi }_l;{\mathrm{\Omega }}_j:=\mathrm{\Psi }\left({\mathrm{\Delta }}_j^{\mathrm{\prime }}\right),L_{R_1}^j:=L_{R_1}\cap {\mathrm{\Omega }}_j;\mathrm{\Omega }=\bigcup _{j=1}^l{\mathrm{\Omega }}_j.\end{array}$For simplicity of calculations, we can restrict ourselves by considering one point on the boundary. So, let the weight function $h\left(z\right)$ be defined as in (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ) for l = 1 and we put: ${\gamma }_1=:\gamma$. To estimate $A_n\left(z\right)$, multiplying the numerator and denominator of the integrand by $h^{\frac{1}{p}}\left(\zeta \right)$ and applying the Hölder inequality, we obtain: $\begin{array}{rl}{A}_n\left(z\right)& ={\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m}\le {\left({\int }_{L_{R_1}}h\right(\zeta \left){\left|P_n\left(\zeta \right)\right|}^p\left|\mathrm{d}\zeta \right|\right)}^{\frac{1}{p}}\\ & \times {\left({\int }_{L_{R_1}}\frac{\left|\mathrm{d}\zeta \right|}{h^{\frac{q}{p}}\left(\zeta \right){|\zeta -z|}^{qm}}\right)}^{\frac{1}{q}}.\end{array}$Using Lemma 5.3 and after replacing the variable $\tau =\mathrm{\Phi }\left(\zeta \right)$, we get: $A_n\left(z\right)⪯{\Vert P_n\Vert }_p\times {\left({\int }_{\left|\tau \right|=R_1}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}}.$To estimate the last integral, we put: (22) $\begin{array}{rl}{E}_{R_1}^{11}& :=\{\tau :\tau \in F_{R_1}^1,|\tau -w_1|<c_1(R_1-1\left)\right\},\\ E_{R_1}^{12}& :=\{\tau :\tau \in F_{R_1}^1,c_1(R_1-1)\le |\tau -w_1|<\eta \},\\ E_{R_1}^{13}& :=\{\tau :\tau \in \mathrm{\Phi }(L_{R_1}),|\tau -w_1|\ge \eta \},\end{array}$(22) where $0<c_1<\eta$ chosen so that $\{\tau :|\tau -w_1|<c_1(R_1-1\left)\right\}\cap \mathrm{\Delta }\ne \varnothing$ and $\mathrm{\Phi }\left(L_{R_1}\right)=\bigcup _{k=1}^3{E}_{R_1}^{1k}$. Then, from (22(22) $\begin{array}{rl}{E}_{R_1}^{11}& :=\{\tau :\tau \in F_{R_1}^1,|\tau -w_1|<c_1(R_1-1\left)\right\},\\ E_{R_1}^{12}& :=\{\tau :\tau \in F_{R_1}^1,c_1(R_1-1)\le |\tau -w_1|<\eta \},\\ E_{R_1}^{13}& :=\{\tau :\tau \in \mathrm{\Phi }(L_{R_1}),|\tau -w_1|\ge \eta \},\end{array}$(22) ) we have: (23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) For any $k=1,2,$ denote by (24) $\begin{array}{rl}{E}_{R_1,1}^{1k}& :=\{\tau \in E_{R_1}^{1k}:\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|\ge \left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|\},E_{R_1,2}^{1k}:=E_{R_1}^{1k}\mathrm{\setminus }{E}_{R_1,1}^{1k},\\ {\left(I\right(E_{R_1,1}^{1k}\left)\right)}^q& :=\{\begin{array}{ll}{\displaystyle {\int }_{E_{R_1,1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{\gamma (q-1)+qm}},}& \mathrm{i}\mathrm{f}\gamma \ge 0,\\ {\displaystyle {\int }_{E_{R_1,1}^{1k}}\frac{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}},}& \mathrm{i}\mathrm{f}\gamma <0,\end{array}\\ {\left(I\right(E_{R_1,2}^{1k}\left)\right)}^q& :={\int }_{E_{R_1,2}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)+qm}},\end{array}$(24) and estimate these integrals separately.

1. Let $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}\le \alpha \le 1$.

1.1. Let $\gamma \ge 0.$ If $z\in \mathrm{\Omega }\left(\delta \right)$, applying Lemma 5.2 to (24(24) $\begin{array}{rl}{E}_{R_1,1}^{1k}& :=\{\tau \in E_{R_1}^{1k}:\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|\ge \left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|\},E_{R_1,2}^{1k}:=E_{R_1}^{1k}\mathrm{\setminus }{E}_{R_1,1}^{1k},\\ {\left(I\right(E_{R_1,1}^{1k}\left)\right)}^q& :=\{\begin{array}{ll}{\displaystyle {\int }_{E_{R_1,1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{\gamma (q-1)+qm}},}& \mathrm{i}\mathrm{f}\gamma \ge 0,\\ {\displaystyle {\int }_{E_{R_1,1}^{1k}}\frac{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}},}& \mathrm{i}\mathrm{f}\gamma <0,\end{array}\\ {\left(I\right(E_{R_1,2}^{1k}\left)\right)}^q& :={\int }_{E_{R_1,2}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)+qm}},\end{array}$(24) ), we get: (25) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{11}\left)\right)}^q& ⪯{\int }_{E_{R_1,1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{\gamma (q-1)+qm}}\\ & ⪯n{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }}}⪯n^{1+\frac{\gamma (q-1)+qm-1}{\alpha }}mes{E}_{R_1,1}^{11}⪯n^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }};\\ I\left(E_{R_1,1}^{11}\right)& ⪯n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}.\\ {\left(I\right(E_{R_1,2}^{11}\left)\right)}^q& ⪯n{\int }_{E_{R_1,2}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }}}⪯n^{1+\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }}mes{E}_{R_1,2}^{11}⪯n^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }};\\ I\left(E_{R_1,2}^{11}\right)& ⪯n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}.\end{array}$(25) (26) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{12}\left)\right)}^q& ⪯n{\int }_{E_{R_1,1}^{12}}\frac{\left|\mathrm{d}\tau \right|}{|\tau -w|\frac{{}^{\gamma (q-1)+qm-1}}{\alpha }}\\ & ⪯n\{\begin{array}{ll}{\displaystyle n^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }-1},}& \gamma (q-1)+qm-1>\alpha ;\\ \ln n,& \gamma (q-1)+qm-1=\alpha ,\\ 1,& \gamma (q-1)+qm-1<\alpha ;\end{array}\\ I\left(E_{R_1,1}^{12}\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }},}& \gamma >-1,& m\ge 2,\\ {\displaystyle n^{\frac{\gamma +1}{\alpha p}},}& p<1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma +1}{\alpha }}.& m=1.\end{array}\end{array}$(26) (27) $\begin{array}{rl}{\left(I\right(E_{R_1,2}^{12}\left)\right)}^q& ⪯n{\int }_{E_{R_1,2}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\gamma (q-1)+qm-1}}\\ & ⪯n\{\begin{array}{ll}{\displaystyle n^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }-1},}& \gamma (q-1)+qm-1>\alpha ,\\ \ln n,& \gamma (q-1)+qm-1=\alpha ,\\ 1,& \gamma (q-1)+qm-1<\alpha ;\end{array}\\ I\left(E_{R_1,2}^{12}\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }},}& \gamma >-1,& m\ge 2,\\ {\displaystyle n^{\frac{\gamma +1}{\alpha p}},}& p<1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma +1}{\alpha }}.& m=1.\end{array}\end{array}$(27) For $\tau \in E_{R_1}^{13}$, we have $\eta <|\tau -w_1|<2\pi R_1$, $|\tau -w|\ge \eta -c_1$. Then, $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪰1$, by Lemma 5.1 and for $|\tau -w_1|\ge \eta$, $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|⪰|\tau -w{|}^{\frac{1}{\alpha }}$, by Lemma 5.2. Applying (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), for $w\in \mathrm{\Delta }(w_1,\eta )$, we get: (28) $\begin{array}{rl}{\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}⪯n{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{[qm-1]}{\alpha }}}\\ & ⪯\{\begin{array}{lll}{\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,& m\ge 2,\\ {\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,& m=1,\\ n\ln n,& qm-1=\alpha ,& m=1,\\ n,& qm-1<\alpha ,& m=1;\end{array}\\ J_2^3\left(z\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{\frac{1}{\alpha }(m-1+\frac{1}{p})},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{p\alpha }},}& p<1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1,\end{array}z\in \mathrm{\Omega }\left(\delta \right).\end{array}$(28) Combining (25(25) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{11}\left)\right)}^q& ⪯{\int }_{E_{R_1,1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{\gamma (q-1)+qm}}\\ & ⪯n{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }}}⪯n^{1+\frac{\gamma (q-1)+qm-1}{\alpha }}mes{E}_{R_1,1}^{11}⪯n^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }};\\ I\left(E_{R_1,1}^{11}\right)& ⪯n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}.\\ {\left(I\right(E_{R_1,2}^{11}\left)\right)}^q& ⪯n{\int }_{E_{R_1,2}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }}}⪯n^{1+\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }}mes{E}_{R_1,2}^{11}⪯n^{\frac{\left[\gamma \right(q-1)+qm-1]}{\alpha }};\\ I\left(E_{R_1,2}^{11}\right)& ⪯n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}.\end{array}$(25) –28(28) $\begin{array}{rl}{\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}⪯n{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{[qm-1]}{\alpha }}}\\ & ⪯\{\begin{array}{lll}{\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,& m\ge 2,\\ {\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,& m=1,\\ n\ln n,& qm-1=\alpha ,& m=1,\\ n,& qm-1<\alpha ,& m=1;\end{array}\\ J_2^3\left(z\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{\frac{1}{\alpha }(m-1+\frac{1}{p})},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{p\alpha }},}& p<1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1,\end{array}z\in \mathrm{\Omega }\left(\delta \right).\end{array}$(28) ), for $p>1,\gamma \ge 0$ and $z\in \mathrm{\Omega }\left(\delta \right)$, we get: (29) $\begin{array}{r}\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}}& \gamma >-1,& m\ge 2,\\ {\displaystyle n^{\frac{\gamma +1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1.\end{array}\end{array}$(29) If $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, then (30) $\begin{array}{rl}{\left(J_2^1\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){|\tau -w_1|}^{\frac{\gamma (q-1)}{\alpha }}}⪯n{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}⪯n^{1+\frac{\gamma (q-1)-1}{\alpha }}mes{E}_{R_1}^{11}\\ & ⪯n^{\frac{\gamma (q-1)-1}{\alpha }};\\ J_2^1\left(z\right)& ⪯n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }}.\\ {\left(J_2^2\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{12}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){|\tau -w_1|}^{\frac{\gamma (q-1)}{\alpha }}}⪯n{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}\\ & ⪯n\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma (q-1)-1}{\alpha }-1},}& \gamma (q-1)-1>\alpha ,\\ \ln n,& \gamma (q-1)-1=\alpha ,\\ 1,& \gamma (q-1)-1<\alpha ;\end{array}\\ J_2^2\left(z\right)& ⪯\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ n^{1-\frac{1}{p}},& p>1+{\displaystyle \frac{\gamma }{1+\alpha }}.\end{array}\\ {\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}}⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n^{(\frac{1}{\alpha }-1)};\\ J_2^3\left(z\right)& ⪯n^{(\frac{1}{\alpha }-1)(1-\frac{1}{p})}.\end{array}$(30) Combining (23(23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) )–(30(30) $\begin{array}{rl}{\left(J_2^1\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){|\tau -w_1|}^{\frac{\gamma (q-1)}{\alpha }}}⪯n{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}⪯n^{1+\frac{\gamma (q-1)-1}{\alpha }}mes{E}_{R_1}^{11}\\ & ⪯n^{\frac{\gamma (q-1)-1}{\alpha }};\\ J_2^1\left(z\right)& ⪯n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }}.\\ {\left(J_2^2\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{12}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){|\tau -w_1|}^{\frac{\gamma (q-1)}{\alpha }}}⪯n{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}\\ & ⪯n\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma (q-1)-1}{\alpha }-1},}& \gamma (q-1)-1>\alpha ,\\ \ln n,& \gamma (q-1)-1=\alpha ,\\ 1,& \gamma (q-1)-1<\alpha ;\end{array}\\ J_2^2\left(z\right)& ⪯\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ n^{1-\frac{1}{p}},& p>1+{\displaystyle \frac{\gamma }{1+\alpha }}.\end{array}\\ {\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}}⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n^{(\frac{1}{\alpha }-1)};\\ J_2^3\left(z\right)& ⪯n^{(\frac{1}{\alpha }-1)(1-\frac{1}{p})}.\end{array}$(30) ), we obtain: (31) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}}& \gamma >-1,& m\ge 2,\\ {\displaystyle n^{\frac{\gamma +1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(31) (32) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(32) 1.2. If $\gamma <0$, for $w\in \mathrm{\Delta }(w_1,\eta )\cap {\mathrm{\Omega }}_R\left(\delta \right)$, such that $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪯\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|$, according Lemma 5.1, we have: (33) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{11}\left)\right)}^q& ⪯{\int }_{E_{R_1,1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\\ & ⪯n{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm-1}{\alpha }}}⪯n^{1+\frac{qm-1}{\alpha }}mes{E}_{R_1}^{11}⪯n^{\frac{qm-1}{\alpha }};\\ I\left(E_{R_1,1}^{11}\right)& ⪯n^{(m-1+\frac{1}{p})\frac{1}{\alpha }}.\end{array}$(33) (34) $\begin{array}{rl}{\left(I\right(E_{R_1,2}^{11}\left)\right)}^q& \approx {\int }_{E_{R_1,2}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)+qm}}⪯n{\int }_{E_{R_1,2}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)+qm-1}{\alpha }}}\\ & ⪯n^{1+\frac{\gamma (q-1)+qm-1}{\alpha }}mes{E}_{R_1,2}^{11}⪯n^{\frac{\gamma (q-1)+qm-1}{\alpha }};\\ I\left(E_{R_1,2}^{11}\right)& ⪯n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}.\end{array}$(34) For $\tau \in E_{R_1}^{12}$ we have $|\tau -w_1|<\eta$ and, so, $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪯1$, from Lemma 5.1. Then, for $w\in \mathrm{\Delta }(w_1,\eta )\cap {\mathrm{\Omega }}_R\left(\delta \right)$, such that $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪯\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|$, applying Lemma 5.2, we get: (35) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{12}\left)\right)}^q& ⪯n{\int }_{E_{R_1,1}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm-1}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& qm-1=\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& qm-1<\alpha ;\end{array}\end{array}$(35) (36) $\begin{array}{rl}I\left(E_{R_1,1}^{12}\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1.\end{array}\\ {\left(I\right(E_{R_1,2}^{12}\left)\right)}^q& ⪯n{\int }_{E_{R_1,2}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm+\gamma (q-1)-1}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{qm+\gamma (q-1)-1}{\alpha }},}& qm+\gamma (q-1)-1>\alpha ,\\ n\ln n,& qm+\gamma (q-1)-1=\alpha ,\\ n,& qm+\gamma (q-1)-1<\alpha ;\end{array}\\ I\left(E_{R_1,2}^{12}\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{\gamma +1}{p\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1.\end{array}\end{array}$(36) For $\tau \in E_{R_1}^{13}$ and each $w\in \mathrm{\Delta }(w_1,\eta )\cap {\mathrm{\Omega }}_R\left(\delta \right)$ we have $\eta <|\tau -w_1|<2\pi \dot{\phantom{a}}{R}_1$. Therefore, from Lemma 5.1 and applying (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: (37) $\begin{array}{rl}{\left(I\right(E_{R_1}^{13}\left)\right)}^q& \approx {\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}⪯n{\int }_{E_{R_1}^{13}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm-1}{\alpha }}}\\ & ⪯\{\begin{array}{ll}{\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& qm-1=\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& qm-1<\alpha ;\end{array}\\ I\left(E_{R_1}^{13}\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1.\end{array}\end{array}$(37) Further, combining (33(33) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{11}\left)\right)}^q& ⪯{\int }_{E_{R_1,1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\\ & ⪯n{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm-1}{\alpha }}}⪯n^{1+\frac{qm-1}{\alpha }}mes{E}_{R_1}^{11}⪯n^{\frac{qm-1}{\alpha }};\\ I\left(E_{R_1,1}^{11}\right)& ⪯n^{(m-1+\frac{1}{p})\frac{1}{\alpha }}.\end{array}$(33) )–(37(37) $\begin{array}{rl}{\left(I\right(E_{R_1}^{13}\left)\right)}^q& \approx {\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}⪯n{\int }_{E_{R_1}^{13}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm-1}{\alpha }}}\\ & ⪯\{\begin{array}{ll}{\displaystyle n^{\frac{qm-1}{\alpha }},}& qm-1>\alpha ,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& qm-1=\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& qm-1<\alpha ;\end{array}\\ I\left(E_{R_1}^{13}\right)& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1.\end{array}\end{array}$(37) ) in case of $\gamma <0$, for $z\in \mathrm{\Omega }\left(\delta \right)$, we have: (38) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+\frac{1}{\alpha },& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+\frac{1}{\alpha },& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+\frac{1}{\alpha },& m=1.\end{array}$(38) If $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, then $|w-w_1|\ge \eta$ and from Lemma 5.2 and (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: $\begin{array}{rl}{\left(J_2^1\right(z\left)\right)}^q& ⪯n{\int }_{E_{R_1}^{11}}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)+1}\left|\mathrm{d}\tau \right|⪯n.mes{E}_{R_1}^{11}⪯1;\\ J_2^1\left(z\right)& ⪯1.\\ {\left(J_2^2\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{12}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\left|\mathrm{d}\tau \right|⪯{\int }_{E_{R_1}^{12}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n;\\ J_2^2\left(z\right)& ⪯n^{1-\frac{1}{p}}.\\ {\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}}⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{\left|\tau \right|-1}⪯n;\\ J_2^3\left(z\right)& ⪯n^{1-\frac{1}{p}}.\end{array}$Combining the last three estimates, in case of $\gamma <0$, for $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, we have: (39) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{1-\frac{1}{p}}.$(39) Then, for $\gamma <0$, from (38(38) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+\frac{1}{\alpha },& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+\frac{1}{\alpha },& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+\frac{1}{\alpha },& m=1.\end{array}$(38) –39(39) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{1-\frac{1}{p}}.$(39) ), we obtain: $\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{llll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& m\ge 1,& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}$and, consequently, for $-1<\gamma <0$, from (23(23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) ), we have: (40) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& m\ge 1& z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}\end{array}$(40) Therefore, combining (29(29) $\begin{array}{r}\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+m-1)\frac{1}{\alpha }}}& \gamma >-1,& m\ge 2,\\ {\displaystyle n^{\frac{\gamma +1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma +1}{\alpha }},& m=1.\end{array}\end{array}$(29) ) and (40(40) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{1}{p}+m-1)\frac{1}{\alpha }},}& p>1,& m\ge 2,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{1}{\alpha }},& m=1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& m\ge 1& z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}\end{array}$(40) ), for any $\gamma >-1$, p>1, we obtain: (41) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(41) (42) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(42) (B) Now, we begin to estimate the $\left|\right({\mathrm{\Phi }}^{-n-1}\left(z\right){)}^{\left(j\right)}|$.

The Cauchy integral representation for the region ${\mathrm{\Omega }}_R$ gives: ${\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}=-\frac{1}{2\pi i}{\int }_{L_{R_1}}\frac{1}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\frac{\mathrm{d}\zeta }{{(\zeta -z)}^{j+1}},z\in {\mathrm{\Omega }}_R.$Then, replacing the variable $\tau =\mathrm{\Phi }\left(\zeta \right)$ and according to (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we obtain: (43) $\begin{array}{rl}\left|{\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\frac{1}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{j+1}}\le \frac{1}{2\pi }{\int }_{L_{R_1}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{j+1}}\\ & ⪯{\int }_{\left|\tau \right|=R_1}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\frac{\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{j+1}}\\ & ⪯n{\int }_{\left|\tau \right|=R_1}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{j}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{j}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}j=\overline{1,m}.\end{array}$(43) Combining estimates (19(19) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\{\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|+\sum _{j=1}^m{C}_m^j\left|{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}\right|\left|P_n^{(m-j)}\left(z\right)\right|\}.$(19) )–(23(23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) ), (41(41) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(41) ), (42(42) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(42) ) and (43(43) $\begin{array}{rl}\left|{\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\frac{1}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{j+1}}\le \frac{1}{2\pi }{\int }_{L_{R_1}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{j+1}}\\ & ⪯{\int }_{\left|\tau \right|=R_1}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\frac{\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{j+1}}\\ & ⪯n{\int }_{\left|\tau \right|=R_1}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{j}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{j}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}j=\overline{1,m}.\end{array}$(43) ), for any $\gamma >-1$, p>1, $m\ge 1$, we get: (44) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{A_n\left(z\right)}{\mathrm{d}(z,L)}+\sum _{j=1}^m{C}_m^j\left|P_n^{(m-j)}\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{j}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(44) where $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ A_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$Therefore, the proof of Theorem 2.1 is completed.

Now, we begin the proof of Theorem 2.2. Note that the proof of Theorem 2.2 is carried out similarly to the proof of Theorem 2.1, supplementing it with appropriate estimates for the given case.

2. Let $L\in Q_{\alpha }^{\text{β}}$ for some $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$.

2.1. Let $\gamma >0$. Since $L\in Q_{\alpha }^{\text{β}}$, for arbitrary $z^{\ast }\in L$ such that $\mathrm{d}\left(\mathrm{\Psi }\right(\tau ),L)=\left|\mathrm{\Psi }\right(\tau )-z^{\ast }|$ and $w^{\ast }:=\mathrm{\Phi }\left(z^{\ast }\right)$, we have $\mathrm{d}\left(\mathrm{\Psi }\right(\tau ),L)=\left|\mathrm{\Psi }\right(\tau )-z^{\ast }|⪯|\tau -w^{\ast }{|}^{\text{β}}$. If $z\in \mathrm{\Omega }\left(\delta \right)$, applying Lemmas 5.1 and 5.2 to (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{11}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{\gamma (q-1)+qm}{\alpha }}}⪯n^{1-\text{β}+\frac{\gamma (q-1)+qm}{\alpha }}mes{E}_{R_1,1}^{11}⪯n^{\frac{\gamma (q-1)+qm-}{\alpha }-\text{β}};\\ I\left(E_{R_1,1}^{11}\right)& ⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}.\\ {\left(I\right(E_{R_1,2}^{11}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,2}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)+qm}{\alpha }}}⪯n^{1-\text{β}+\frac{\gamma (q-1)+qm}{\alpha }}mes{E}_{R_1,2}^{11}⪯n^{\frac{\gamma (q-1)+qm-}{\alpha }-\text{β}};\\ I\left(E_{R_1,2}^{11}\right)& ⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},\end{array}$or (45) $I\left(E_{R_1,1}^{11}\right)+I\left(E_{R_1,2}^{11}\right)⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}.$(45) Further, (46) $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{12}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,1}^{12}}\frac{\left|\mathrm{d}\tau \right|}{|\tau -w|\frac{{}^{\gamma (q-1)+qm}}{\alpha }}⪯n^{1-\text{β}}\cdot n^{\frac{\gamma (q-1)+qm}{\alpha }-1}=n^{\frac{\gamma (q-1)+qm}{\alpha }-\text{β}};\\ I\left(E_{R_1,1}^{12}\right)& ⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}m\ge 1,\end{array}$(46) and (47) $\begin{array}{rl}{\left(I\right(E_{R_1,2}^{12}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,2}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\gamma (q-1)+qm}}⪯n^{1-\text{β}}\cdot n^{\frac{\gamma (q-1)+qm}{\alpha }-1}=n^{\frac{\gamma (q-1)+qm}{\alpha }-\text{β}};\\ I\left(E_{R_1,2}^{12}\right)& ⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}m\ge 1.\end{array}$(47) For $\tau \in E_{R_1}^{13}$ we have $\eta <|\tau -w_1|<2\pi \dot{\phantom{a}}{R}_1$ and $|\tau -w|\ge \eta -c_1$. Therefore, $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪰1$, by Lemma 5.1 and $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|⪰|\tau -w{|}^{\frac{1}{\alpha }}$, for $|\tau -w_1|\ge \eta$, by Lemmas 5.1 and 5.2. Then, for $w\in \mathrm{\Delta }(w_1,\eta )$, applying (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: (48) ${\left(J_2^3\right(z\left)\right)}^q⪯n^{1-\text{β}}{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm}{\alpha }}}⪯n^{\frac{qm}{\alpha }-\text{β}};J_2^3\left(z\right)⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}},z\in \mathrm{\Omega }\left(\delta \right).$(48) Combining (45(45) $I\left(E_{R_1,1}^{11}\right)+I\left(E_{R_1,2}^{11}\right)⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}.$(45) –48(48) ${\left(J_2^3\right(z\left)\right)}^q⪯n^{1-\text{β}}{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm}{\alpha }}}⪯n^{\frac{qm}{\alpha }-\text{β}};J_2^3\left(z\right)⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}},z\in \mathrm{\Omega }\left(\delta \right).$(48) ), for $p>1,\gamma >0$ and $z\in \mathrm{\Omega }\left(\delta \right)$, we get: (49) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}.$(49) If $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, then $\begin{array}{rl}{\left(J_2^1\right(z\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)}{\alpha }}}\\ & ⪯n^{1-\text{β}+\frac{\gamma (q-1)}{\alpha }}mes{E}_{R_1}^{11}⪯n^{\frac{\gamma (q-1)}{\alpha }-\text{β}};J_2^1\left(z\right)⪯n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}}.\\ {\left(J_2^2\right(z\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma (q-1)}{\alpha }-\text{β}},}& \gamma (q-1)>\alpha ,\\ n^{1-\text{β}}\ln n,& \gamma (q-1)=\alpha ,\\ n^{1-\text{β}},& \gamma (q-1)<\alpha ;\end{array}\\ J_2^2\left(z\right)& ⪯\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }}.\end{array}\\ {\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n^{1-\text{β}};J_2^3\left(z\right)⪯n^{(1-\text{β})(1-\frac{1}{p})}.\end{array}$Therefore, for $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, we get: (50) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},\end{array}$(50) and, for any $p>1,\gamma >0$, from (23(23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) ), (49(49) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}.$(49) ) and (50(50) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},\end{array}$(50) ), we obtain: (51) $\begin{array}{rl}{A}_n\left(z\right)& ⪯n^{(\frac{\gamma }{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(51) (52) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(52) 2.2. If $\gamma \le 0$, for$w\in \mathrm{\Delta }(w_1,\eta )\cap {\mathrm{\Omega }}_R\left(\delta \right)$, such that $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪯\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|$, according to Lemma 5.1, we have: $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{11}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm}{\alpha }}}⪯n^{1-\text{β}+\frac{qm}{\alpha }}mes{E}_{R_1}^{11}⪯n^{\frac{qm}{\alpha }-\text{β}};\\ I\left(E_{R_1,1}^{11}\right)& ⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}.\\ {\left(I\right(E_{R_1,2}^{11}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,2}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)+qm}{\alpha }}}⪯n^{1-\text{β}+\frac{\gamma (q-1)+qm}{\alpha }}mes{E}_{R_1,2}^{11}⪯n^{\frac{\gamma (q-1)+qm}{\alpha }-\text{β}};\\ I\left(E_{R_1,2}^{11}\right)& ⪯n^{\frac{\gamma +pm}{p\alpha }-(1-\frac{1}{p})\text{β}}.\end{array}$For $\tau \in E_{R_1}^{12}|\tau -w_1|<\eta$ holds and, therefore, $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪯1$, by Lemma 5.1. Then, for $w\in \mathrm{\Delta }(w_1,\eta )\cap {\mathrm{\Omega }}_R\left(\delta \right)$, such that $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪯\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|$, applying Lemma 5.2, we get: $\begin{array}{rl}{\left(I\right(E_{R_1,1}^{12}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,1}^{12}}\frac{{|\tau -w_1|}^{(-\gamma )(q-1)\text{β}}\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm}{\alpha }}}⪯n^{\frac{qm}{\alpha }-\text{β}};I\left(E_{R_1,1}^{12}\right)⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}.\\ {\left(I\right(E_{R_1,2}^{12}\left)\right)}^q& ⪯n^{1-\text{β}}{\int }_{E_{R_1,2}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm}{\alpha }}}⪯n^{\frac{qm}{\alpha }-\text{β}};I\left(E_{R_1,2}^{12}\right)⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}.\end{array}$For $\tau \in E_{R_1}^{13}$ and each $w\in \mathrm{\Delta }(w_1,\eta )\cap {\mathrm{\Omega }}_R\left(\delta \right)$, $\eta <|\tau -w_1|<2\pi \dot{\phantom{a}}{R}_1$ holds. Therefore, from Lemma 5.1 and applying (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: $\begin{array}{rl}{\left(I\right(E_{R_1}^{13}\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}⪯n^{1-\text{β}}{\int }_{E_{R_1}^{13}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{qm}{\alpha }}}⪯n^{\frac{qm}{\alpha }-\text{β}};\\ I\left(E_{R_1}^{13}\right)& ⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}.\end{array}$Therefore, case of $\gamma \le 0$ for $z\in \mathrm{\Omega }\left(\delta \right)$, we have: (53) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}.$(53) If $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, then $|w-w_1|\ge \eta$, from Lemma 5.2 and from (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: $\begin{array}{rl}{\left(J_2^1\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|\\ & ⪯n^{1-\text{β}}{\int }_{E_{R_1}^{11}}\left|\mathrm{d}\tau \right|⪯n^{1-\text{β}}.mes{E}_{R_1}^{11}⪯n^{-\text{β}}⪯1.\\ J_2^1\left(z\right)& ⪯1.\\ {\left(J_2^2\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{12}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\left|\mathrm{d}\tau \right|\\ & ⪯{\int }_{E_{R_1}^{12}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n^{1-\text{β}};\\ J_2^2\left(z\right)& ⪯n^{(1-\frac{1}{p})(1-\text{β})}.\\ {\left(J_2^3\right(z\left)\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}}⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n^{1-\text{β}};\\ J_2^3\left(z\right)& ⪯n^{(1-\frac{1}{p})(1-\text{β})}.\end{array}$Combining the last tree estimates, in case of $\gamma \le 0$ for $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$, we have: (54) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{(1-\frac{1}{p})(1-\text{β})}.$(54) Then, for $\gamma \le 0$, from (53(53) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}.$(53) ) and (54(54) $\sum _{k=1}^3{J}_n^k\left(z\right)⪯n^{(1-\frac{1}{p})(1-\text{β})}.$(54) ), we obtain: $\sum _{k=1}^3{J}_n^k\left(z\right)⪯\{\begin{array}{lll}{\displaystyle n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}}& p>1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1,& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}$and, consequently, for $-1<\gamma \le 0$ from (23(23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) ), we have: (55) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\cdot \{\begin{array}{lll}{\displaystyle n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}}& p>1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1,& z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}\end{array}$(55) Therefore, combining (52(52) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(52) ) and (55(55) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\cdot \{\begin{array}{lll}{\displaystyle n^{\frac{m}{\alpha }-(1-\frac{1}{p})\text{β}}}& p>1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1,& z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}\end{array}$(55) ), for any $\gamma >-1$, p>1, we get: (56) $A_n\left(z\right)⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{{\gamma }^{\ast }}{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& p>1,& \gamma >-1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}$(56) where ${\gamma }^{\ast }:=max\{0;\gamma \}$.

In this case for the estimate $\left|\right({\mathrm{\Phi }}^{-n-1}\left(z\right){)}^{\left(j\right)}|$, according to (43(43) $\begin{array}{rl}\left|{\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\frac{1}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{j+1}}\le \frac{1}{2\pi }{\int }_{L_{R_1}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{j+1}}\\ & ⪯{\int }_{\left|\tau \right|=R_1}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\frac{\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{j+1}}\\ & ⪯n{\int }_{\left|\tau \right|=R_1}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{j}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{j}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}j=\overline{1,m}.\end{array}$(43) ), we obtain: (57) $\left|{\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}\right|⪯n^{1-\text{β}}{\int }_{\left|\tau \right|=R_1}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{j+1}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{j+1}{\alpha }-\text{β}},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n^{1-\text{β}},& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}j=\overline{1,m}.$(57) Then combining estimates (19(19) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\{\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|+\sum _{j=1}^m{C}_m^j\left|{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}\right|\left|P_n^{(m-j)}\left(z\right)\right|\}.$(19) )–(23(23) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\times \sum _{k=1}^3{J}_n^k\left(z\right),\\ J_n^k\left(z\right)& :={\left({\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w\left)\right|}^{qm}}\right)}^{\frac{1}{q}},k=1,2,3.\end{array}$(23) ), (41(41) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(41) ), (42(42) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(42) ) and (56(56) $A_n\left(z\right)⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{{\gamma }^{\ast }}{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& p>1,& \gamma >-1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}$(56) ), we get: (58) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{A_n\left(z\right)}{\mathrm{d}(z,L)}+\sum _{j=1}^m{C}_m^j\left|P_n^{(m-j)}\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{j+1}{\alpha }-\text{β}},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n^{1-\text{β}},& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(58) where for any $\gamma >-1,p>1,m\ge 1$ $A_n\left(z\right)⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{{\gamma }^{\ast }}{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& p>1,& \gamma >-1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{\alpha }},& \gamma >0,& z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$The proof of Theorem 2.2 is completed.

Proof

Proof of Theorem 2.3

Now let's start evaluations of $\left|P_n\right(z\left)\right|$. Proceeding in the same way as at the beginning of the proof of Theorems 2.1 and 2.2 for estimates (20(20) $\begin{array}{rl}\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{m+1}}\\ & \le \frac{1}{2\pi \mathrm{d}(z,L_{R_1})}{\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m}.\end{array}$(20) ) and (21(21) $A_n\left(z\right):={\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m},$(21) ), we have: (59) $\left|P_n\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p(J_{n,2}^1+J_{n,2}^2+J_{n,2}^3),$(59) where ${\left(J_{n,2}^k\right)}^q:={\int }_{E_{R_1}^{1k}}\frac{\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|}{{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}},k=1,2,3.$So, for any k = 1, 2, 3 we will estimate the $J_{n,2}^k$  for case $L\in {\tilde{Q}}_{\alpha }$.

Let $L\in {\tilde{Q}}_{\alpha }$, for some $\frac{1}{2}\le \alpha \le 1$.

1. Let $\gamma >0$. Applying Lemma 5.2 and (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: (60) $\begin{array}{rl}{\left(J_{n,2}^1\right)}^q& ⪯{\int }_{E_{R_1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}}\\ & ⪯n{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}⪯n^{1+\frac{\gamma (q-1)-1}{\alpha }}mes{E}_{R_1}^{11}\\ & ⪯n^{\frac{\gamma (q-1)-1}{\alpha }};\end{array}$(60) (61) $\begin{array}{rl}{J}_{n,2}^1& ⪯n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }}.{\left(J_{n,2}^2\right)}^q\\ & ⪯n{\int }_{E_{R_1}^{12}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}⪯n\{\begin{array}{ll}{\displaystyle n^{\frac{\gamma (q-1)-1}{\alpha }},}& \gamma (q-1)-1>\alpha ,\\ \ln n,& \gamma (q-1)-1=\alpha ,\\ 1,& \gamma (q-1)-1<\alpha ;\end{array}\\ J_{n,2}^2& ⪯\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\end{array}\end{array}$(61) For $\tau \in E_{R_1}^{13}$,  we have $\eta <|\tau -w_1|<2\pi \dot{\phantom{a}}{R}_1$. Therefore, $\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|⪰1$, by Lemma 5.1 and applying (18(18) $\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\approx \left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|(\left|\tau \right|-1).$(18) ), we get: (62) $\begin{array}{rl}{\left(J_{n,2}^3\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n;J_{n,2}^3⪯n^{1-\frac{1}{p}}.\end{array}$(62) Combining (60(60) $\begin{array}{rl}{\left(J_{n,2}^1\right)}^q& ⪯{\int }_{E_{R_1}^{11}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)\left|\mathrm{d}\tau \right|}{(\left|\tau \right|-1){\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{\gamma (q-1)}}\\ & ⪯n{\int }_{E_{R_1}^{11}}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w_1|}^{\frac{\gamma (q-1)-1}{\alpha }}}⪯n^{1+\frac{\gamma (q-1)-1}{\alpha }}mes{E}_{R_1}^{11}\\ & ⪯n^{\frac{\gamma (q-1)-1}{\alpha }};\end{array}$(60) –62(62) $\begin{array}{rl}{\left(J_{n,2}^3\right)}^q& ⪯{\int }_{E_{R_1}^{13}}\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\left|\mathrm{d}\tau \right|⪯{\int }_{E_{R_1}^{13}}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n;J_{n,2}^3⪯n^{1-\frac{1}{p}}.\end{array}$(62) ), for $p>1,\gamma >0$ and $z\in {\mathrm{\Omega }}_R$, we get: (63) $\begin{array}{rlr}\sum _{k=1}^3{J}_{n,2}^k& ⪯& \{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& \gamma \le 1+\alpha .\end{array}\end{array}$(63) From (59(59) $\left|P_n\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p(J_{n,2}^1+J_{n,2}^2+J_{n,2}^3),$(59) )–(63(63) $\begin{array}{rlr}\sum _{k=1}^3{J}_{n,2}^k& ⪯& \{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& \gamma \le 1+\alpha .\end{array}\end{array}$(63) ), we obtain: (64) $\left|P_n\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& \gamma \le 1+\alpha .\end{array}$(64) 2. If $\gamma \le 0$, analogously we have: $\begin{array}{rl}{\left(J_{n,2}^1\right)}^q& ⪯n{\int }_{E_{R_1}^{11}}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)+1}\left|\mathrm{d}\tau \right|\\ & ⪯n{\int }_{E_{R_1}^{11}}\left|\mathrm{d}\tau \right|⪯n\cdot mes{E}_{R_1}^{11}⪯1;J_{n,2}^1⪯1.\\ {\left(J_{n,2}^2\right)}^q& ⪯{\int }_{E_{R_1}^{12}}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n;J_{n,2}^2⪯n^{1-\frac{1}{p}}.\\ {\left(J_{n,2}^3\right)}^q& ⪯{\int }_{E_{R_1}^{13}}{\left|\mathrm{\Psi }\right(\tau )-\mathrm{\Psi }(w_1\left)\right|}^{(-\gamma )(q-1)}\frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}\left|\mathrm{d}\tau \right|⪯n;J_{n,2}^3⪯n^{1-\frac{1}{p}}.\end{array}$Therefore, in case of $\gamma \le 0$ for $z\in {\mathrm{\Omega }}_R$, we have: (65) $\sum _{k=1}^3{J}_{n,2}^k⪯n^{1-\frac{1}{p}}.$(65) Combining estimates (19(19) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\{\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|+\sum _{j=1}^m{C}_m^j\left|{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}\right|\left|P_n^{(m-j)}\left(z\right)\right|\}.$(19) )–(59(59) $\left|P_n\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p(J_{n,2}^1+J_{n,2}^2+J_{n,2}^3),$(59) ), (65(65) $\sum _{k=1}^3{J}_{n,2}^k⪯n^{1-\frac{1}{p}}.$(65) ), we get: $\begin{array}{rl}\left|P_n\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1,& -1<\gamma \le 1+\alpha ,\end{array}\end{array}$and, we complete the proof of Theorem 2.3.

Proofs

Proofs of Theorems 2.5 and 2.6

From (41(41) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(41) ), (42(42) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(42) ) and (44(44) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{A_n\left(z\right)}{\mathrm{d}(z,L)}+\sum _{j=1}^m{C}_m^j\left|P_n^{(m-j)}\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{j}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(44) ), we get: (66) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}\left[A_n\right(z)+\left|P_n\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{1}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(66) where for any $\gamma >-1,p>1,m=1$ $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ A_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right);\\ {\gamma }^{\ast }& :=max\{0;\gamma \}.\end{array}$Taking into account estimates for $\left|P_n\right(z\left)\right|$ from (10(10) $\left|P_n\left(z\right)\right|\le c_3\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^3,$(10) ),  (11(11) $\left|P_n\left(z\right)\right|\le c_4\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^3,$(11) ) respectively and combining with (66(66) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}\left[A_n\right(z)+\left|P_n\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{1}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(66) ) prove the needed estimates. $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma +1}{p\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{1}{\alpha }}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{1}{\alpha }},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{1}{\alpha }},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ \left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }+1},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{2-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{2-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >1+\alpha ,\\ {\displaystyle n^{2-\frac{1}{p}},}& p>1,& -1<\gamma \le 1+\alpha ,\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$Analogously for the case of $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$, $\frac{1}{2}\le \alpha \le 1,0<\text{β}\le 1$, from (56(56) $A_n\left(z\right)⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{{\gamma }^{\ast }}{p}+m)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& p>1,& \gamma >-1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}$(56) ), (57(57) $\left|{\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}\right|⪯n^{1-\text{β}}{\int }_{\left|\tau \right|=R_1}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{j+1}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{j+1}{\alpha }-\text{β}},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n^{1-\text{β}},& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}j=\overline{1,m}.$(57) ) and (66(66) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}\left[A_n\right(z)+\left|P_n\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{1}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(66) ), we get: $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}\left[A_n\right(z)+A_{n,p}^3\{\begin{array}{ll}{\displaystyle n^{\frac{2}{\alpha }-\text{β}},}& z\in \mathrm{\Omega }\left(\delta \right),\\ n^{1-\text{β}},& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$where for any $\gamma >-1,p>1,m=1$, $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{(\frac{{\gamma }^{\ast }}{p}+1)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& p>1,& \gamma >-1,& z\in \mathrm{\Omega }\left(\delta \right),\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right),\\ {\displaystyle n^{(1-\frac{1}{p})(1-\text{β})},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{\alpha }},& \gamma >0& z\in \hat{\mathrm{\Omega }}\left(\delta \right);\end{array}\\ A_{n,p}^3& ⪯{\Vert P_n\Vert }_p\{\begin{array}{llll}{\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,& z\in {\mathrm{\Omega }}_R,\\ {\displaystyle {(n^{1-\text{β}}\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,& z\in {\mathrm{\Omega }}_R,\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,& z\in {\mathrm{\Omega }}_R,\\ {\displaystyle n^{(1-\text{β})(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha .& z\in {\mathrm{\Omega }}_R;\end{array}\\ {\gamma }^{\ast }& :=max\{0;\gamma \}.\end{array}$Therefore, (67) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{2}{\alpha }-(2-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha ,\end{array}\end{array}$(67) for $z\in \mathrm{\Omega }\left(\delta \right)$ and (68) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+1-(2-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha ,\end{array}\end{array}$(68) for $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$. Taking into account estimates for $\left|P_n\right(z\left)\right|$ from (10(10) $\left|P_n\left(z\right)\right|\le c_3\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^3,$(10) ), (11(11) $\left|P_n\left(z\right)\right|\le c_4\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p{A}_{n,p}^3,$(11) ), respectively, and combining with (66(66) $\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{\mathrm{d}(z,L)}\left[A_n\right(z)+\left|P_n\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{1}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(66) ) gives the necessary estimates.

Proofs

Proofs of Theorems 2.7 and 2.8

According to the Theorems 2.1, 2.3, 2.5 and estimates (20(20) $\begin{array}{rl}\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{m+1}}\\ & \le \frac{1}{2\pi \mathrm{d}(z,L_{R_1})}{\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m}.\end{array}$(20) ), (21(21) $A_n\left(z\right):={\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m},$(21) ), (41(41) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{lll}{\displaystyle n^{(\frac{{\gamma }^{\ast }+1}{p}+m-1)\frac{1}{\alpha }}}& p>1,& m\ge 2,\\ {\displaystyle n^{\frac{{\gamma }^{\ast }+1}{\alpha p}},}& 1<p<1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }+1}{\alpha }},& m=1,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\end{array}$(41) ), (42(42) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle {(n\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},\\ {\displaystyle n^{1-\frac{1}{p}},}& p>1+{\displaystyle \frac{{\gamma }^{\ast }}{1+\alpha }},\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$(42) ), (44(44) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{A_n\left(z\right)}{\mathrm{d}(z,L)}+\sum _{j=1}^m{C}_m^j\left|P_n^{(m-j)}\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{j}{\alpha }},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n,& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(44) ), for $L\in {\tilde{Q}}_{\alpha }$, $\frac{1}{2}\le \alpha \le 1$, m = 2 and p>1 from (19(19) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\{\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|+\sum _{j=1}^m{C}_m^j\left|{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(j\right)}\right|\left|P_n^{(m-j)}\left(z\right)\right|\}.$(19) ), we have: $\begin{array}{rl}\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|& \le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\mathrm{\prime }\mathrm{\prime }}\right|+C_2^1{B}_{n,1}^1\left|P_n^{\mathrm{\prime }}\left(z\right)\right|+C_2^2{B}_{n,2}^1\left|P_n\left(z\right)\right|]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|\left[\frac{{\Vert P_n\Vert }_p}{\mathrm{d}(z,L)}{A}_n^1\right(z,2)+C_2^1{B}_{n,1}^1\left|P_n^{\mathrm{\prime }}\left(z\right)\right|+C_2^2{B}_{n,2}^1\left|P_n\left(z\right)\right|].\end{array}$Substituting estimates for $B_{n,j}^1$, j = 1, 2, $\left|P_n\right(z\left)\right|$ and $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ from Theorems 2.1, 2.3 and 2.5 respectively, we get: $\begin{array}{rl}\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}+1)\frac{1}{\alpha }},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{2}{\alpha }}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{2}{\alpha }},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{1-\frac{1}{p}+\frac{2}{\alpha }},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right);\\ \left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{(\frac{\gamma +1}{p}-1)\frac{1}{\alpha }+2},}& 1<p<1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{3-\frac{1}{p}}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{3-\frac{1}{p}},}& p>1+{\displaystyle \frac{\gamma }{1+\alpha }},& \gamma >0,\\ {\displaystyle n^{3-\frac{1}{p}},}& p>1,& -1<\gamma \le 0,\end{array}\mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right).\end{array}$Now, according to Theorems 2.2, 2.4, 2.6 and estimates (20(20) $\begin{array}{rl}\left|{\left(\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\left(m\right)}\right|& \le \frac{1}{2\pi }{\int }_{L_{R_1}}\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{m+1}}\\ & \le \frac{1}{2\pi \mathrm{d}(z,L_{R_1})}{\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m}.\end{array}$(20) ), (21(21) $A_n\left(z\right):={\int }_{L_{R_1}}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^m},$(21) ), (58(58) $\left|P_n^{\left(m\right)}\left(z\right)\right|\le \left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{A_n\left(z\right)}{\mathrm{d}(z,L)}+\sum _{j=1}^m{C}_m^j\left|P_n^{(m-j)}\left(z\right)\right|\{\begin{array}{ll}{\displaystyle n^{\frac{j+1}{\alpha }-\text{β}},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n^{1-\text{β}},& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}]$(58) ), (57(57) $\left|{\left({\mathrm{\Phi }}^{-n-1}\right(z\left)\right)}^{\left(j\right)}\right|⪯n^{1-\text{β}}{\int }_{\left|\tau \right|=R_1}\frac{\left|\mathrm{d}\tau \right|}{{|\tau -w|}^{\frac{j+1}{\alpha }}}⪯\{\begin{array}{ll}{\displaystyle n^{\frac{j+1}{\alpha }-\text{β}},}& \mathrm{i}\mathrm{f}z\in \mathrm{\Omega }\left(\delta \right),\\ n^{1-\text{β}},& \mathrm{i}\mathrm{f}z\in \hat{\mathrm{\Omega }}\left(\delta \right),\end{array}j=\overline{1,m}.$(57) ), (67(67) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{2}{\alpha }-(2-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha ,\end{array}\end{array}$(67) ), (68(68) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+1-(2-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})}{(\ln n)}^{1-\frac{1}{p}},}& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{1-\text{β}(2-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le \alpha ,\end{array}\end{array}$(68) ), for the case $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$, $\frac{1}{2}\le \alpha \le 1$, $0<\text{β}\le 1$, m = 2 and p>1, we have: $\begin{array}{rl}\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{4}{\alpha }-(3-\frac{1}{p})\text{β}},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})}}& & \\ {(\ln n)}^{1-\frac{1}{p}},& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >\alpha ,\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& p>1\ge {\displaystyle \frac{\gamma +\alpha }{2+\alpha -2\alpha \text{β}}},& \begin{array}{l}0<\gamma \le 2(1-\alpha \text{β})\le \alpha ,\\ {\displaystyle \alpha \ge \frac{2}{1+2\text{β}};\text{β}\ge \frac{1}{2},}\end{array}\\ {\displaystyle n^{(\frac{\gamma }{p}+2)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<\frac{\gamma +\alpha }{2+\alpha -2\alpha \text{β}}& \begin{array}{l}2(1-\alpha \text{β})<\gamma \le \alpha ,\\ {\displaystyle \alpha <\frac{2}{1+2\text{β}};\text{β}\ge \frac{1}{2},}\end{array}\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& p\ge \frac{\gamma +\alpha }{2+\alpha -2\alpha \text{β}}>1,& \begin{array}{l}2(1-\alpha \text{β})<\gamma \le \alpha ,\\ {\displaystyle \alpha \ge \frac{2}{1+2\text{β}};\text{β}\ge \frac{1}{2},}\end{array}\\ {\displaystyle n^{\frac{4}{\alpha }-\text{β}(3-\frac{1}{p})+(1-\frac{1}{p})},}& p>1,& -1<\gamma \le 0,\end{array}\end{array}$if $z\in \mathrm{\Omega }\left(\delta \right)$ and $\begin{array}{rl}\left|P_n^{\mathrm{\prime }\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{3(n+1)}\right(z\left)\right|}{\mathrm{d}(z,L_{R_1})}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{lll}{\displaystyle n^{\frac{\gamma }{p\alpha }+\frac{2}{\alpha }-(3-\frac{1}{p})\text{β}+1},}& 1<p<1+{\displaystyle \frac{\gamma }{\alpha }},& \alpha <\gamma \le 2-2\alpha \text{β}+2\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})}}& & \\ {(\ln n)}^{1-\frac{1}{p}},& p=1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p>1+{\displaystyle \frac{\gamma }{\alpha }},& \gamma >0,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p\ge {\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}>1& 0<\gamma \le \alpha ;\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p\ge {\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}>1& \gamma >2-2\alpha \text{β}+2\alpha ,\\ {\displaystyle n^{\frac{\gamma }{p\alpha }-(1-\frac{1}{p})\text{β}},}& 1<p<{\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}& \gamma >2-2\alpha \text{β}+2\alpha ,\\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p>1\ge {\displaystyle \frac{\gamma +\alpha }{2+2\alpha -2\alpha \text{β}}}& 0<\gamma \le \alpha \\ {\displaystyle n^{\frac{2}{\alpha }-\text{β}(3-\frac{1}{p})+(2-\frac{1}{p})},}& p>1,& -1<\gamma \le 0,\end{array}\end{array}$if $z\in \hat{\mathrm{\Omega }}\left(\delta \right)$. Therefore, the proof of Theorems 2.7 and 2.8 is completed.

In conclusion, note that in proofs everywhere there is a quantity $\mathrm{d}(z,L_{R_1})$. Let us show that $\mathrm{d}(z,L_{R_1})⪰\mathrm{d}(z,L)$ holds for all $z\in {\mathrm{\Omega }}_R$. For the points $z\notin \mathrm{\Omega }(L_{R_1},\mathrm{d}(L_{R_1},L_R\left)\right)$, we have: $\mathrm{d}(z,L_{R_1})⪰\delta ⪰\mathrm{d}(z,L)$. Now, let $z\in \mathrm{\Omega }(L_{R_1},\mathrm{d}(L_{R_1},L_R\left)\right)$. Denote by ${\xi }_1\in L_{R_1}$ the point such that $\mathrm{d}(z,L_{R_1})=|z-{\xi }_1|$, and point ${\xi }_2\in L$,such that $\mathrm{d}(z,L)=|z-{\xi }_2|$, and for $w=\mathrm{\Phi }\left(z\right),t_1=\mathrm{\Phi }\left({\xi }_1\right),t_2=\mathrm{\Phi }\left({\xi }_2\right)$, we have: $|w-w_1|\ge \left|\right|w-w_2|-|w_2-w_1\left|\right|\ge \left|\right|w-w_2|-\frac{1}{2}|w-w_2\left|\right|\ge \frac{1}{2}|w-w_2|$. Then, according to Lemma 5.1, we obtain $\mathrm{d}(z,L_{R_1})⪰\mathrm{d}(z,L)$.

Proofs

Proofs of Theorems 3.1 and 3.2

First of all, we give two theorems that we will use in this case, and after then we give an estimate for  $\left|P_n^{\left(m\right)}\right(z\left)\right|$, $z\in \overline{G}$, for each $m\ge 1$.

Theorem A

[15, Th. 2.4]

Assume that $L\in {\tilde{Q}}_{\alpha }$ for some $\frac{1}{2}<\alpha \le 1;P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$ and $h\left(z\right)$ defined as in (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any p>1 there exists a constant $c_9=c_9(L,p,\gamma )>0$ such that the following is fulfilled: (69) ${\Vert P_n\Vert }_{\mathrm{\infty }}\le c_9{n}^{\frac{{\gamma }^{\ast }+1}{p\alpha }}{\Vert P_n\Vert }_p.$(69)

Theorem B

[11, Th.1.5]

Assume that $L\in {\tilde{Q}}_{\alpha }^{\text{β}}$ for some $\frac{1}{2}<\alpha \le 1$ and $0<\text{β}\le 1;P_n\in {\mathrm{\wp }}_n,n\in \mathbb{N}$ and $h\left(z\right)$ defined as in (1(1) $h\left(z\right):=h_0\left(z\right)\prod _{j=1}^l{|z-z_j|}^{{\gamma }_j},$(1) ). Then, for any p>1 there exists a constant $c_{10}=c_{10}(L,p,\gamma )>0$ such that the following is fulfilled: (70) ${\Vert P_n\Vert }_{\mathrm{\infty }}\le c_{10}{n}^{\frac{{\gamma }^{\ast }+p}{\alpha p}-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,$(70) where ${\gamma }^{\ast }$ is defined as in (8(8) ${\gamma }^{\ast }:=max\{0;{\gamma }_k,k=\overline{1,l}\}.$(8) ).

Now, we will give an estimate for $\left|P_n^{\left(m\right)}\right(z\left)\right|$, $z\in \overline{G}$, for each $m\ge 1$ and for the regions ${\tilde{Q}}_{\alpha }$ and ${\tilde{Q}}_{\alpha }^{\text{β}}$, respectively. Let $p>1,U:=B(z,\mathrm{d}(z,L_{R_1}\left)\right)$ and $z\in L$ is an arbitrary fixed point. By the Cauchy integral formulas for mth derivatives, we have: $P_n^{\left(m\right)}\left(z\right)=\frac{m!}{2\pi i}{\int }_{\mathrm{\partial }U}\frac{P_n\left(t\right)}{(t-z{)}^{m+1}}\mathrm{d}t,m=0,1,2,\dots$Then, applying (3(3) ${\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},\mathrm{\forall }{P}_n\in {\mathrm{\wp }}_n.$(3) ), we get: $\begin{array}{rl}\left|P_n^{\left(m\right)}\right(z\left)\right|& \le \frac{m!}{2\pi }\underset{z\in \mathrm{\partial }U}{max}\left|P_n\right(t\left)\right|{\int }_{\mathrm{\partial }U}\frac{\left|\mathrm{d}t\right|}{{|t-z|}^{m+1}}\\ & ⪯\underset{t\in \overline{G_{R_1}}}{max}\left|P_n\right(t\left)\right|\frac{2\pi \mathrm{d}(z,L_R)}{d^{m+1}(z,L_R)}⪯\underset{t\in \overline{G}}{max}\left|P_n\right(t\left)\right|\frac{1}{d^m(z,L_R)}.\end{array}$Now, applying Theorems thmA and thmBand using Corollary 5.1, we get: $\begin{array}{rl}\left|P_n^{\left(m\right)}\right(z\left)\right|& ⪯n^{\frac{{\gamma }^{\ast }+1}{p\alpha }}{\Vert P_n\Vert }_p\cdot n^{\frac{m}{\alpha }}⪯n^{(\frac{{\gamma }^{\ast }+1}{p}+m)\frac{1}{\alpha }}\Vert P_n{\Vert }_p;\\ \left|P_n^{\left(m\right)}\right(z\left)\right|& ⪯n^{\frac{{\gamma }^{\ast }+p}{\alpha p}-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p\cdot n^{\frac{m}{\alpha }}⪯n^{(\frac{{\gamma }^{\ast }}{p}+m+1)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}\Vert P_n\Vert .\end{array}$Since $z\in L$ is arbitrary, we complete the proof of Theorems 3.1 and 3.2.

Proof

Proof of Remark

Sharpness of the inequalities (16(16) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_9{n}^{(\frac{{\gamma }^{\ast }+1}{p}+m)\frac{1}{\alpha }}{\Vert P_n\Vert }_p,$(16) ) and (17(17) ${\Vert P_n^{\left(m\right)}\Vert }_{\mathrm{\infty }}\le c_{10}{n}^{(\frac{{\gamma }^{\ast }}{p}+m+1)\frac{1}{\alpha }-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,$(17) ) can be argued as follows. These inequalities can be interpreted as a combination of the well-known sharp Markov inequalities $\Vert P_n^{\left(m\right)}{\Vert }_{\mathrm{\infty }}⪯n^m\Vert P_n{\Vert }_{\mathrm{\infty }}$ with inequalities (69(69) ${\Vert P_n\Vert }_{\mathrm{\infty }}\le c_9{n}^{\frac{{\gamma }^{\ast }+1}{p\alpha }}{\Vert P_n\Vert }_p.$(69) ) and (70(70) ${\Vert P_n\Vert }_{\mathrm{\infty }}\le c_{10}{n}^{\frac{{\gamma }^{\ast }+p}{\alpha p}-(1-\frac{1}{p})\text{β}}{\Vert P_n\Vert }_p,$(70) ), respectively. And the sharpness of the last inequalities can be verified to the following examples: For the polynomial $T_n\left(z\right)=1+z+\cdots +z^n$, $h^{\ast }\left(z\right)=$ $h_0\left(z\right)$, $h^{\ast \ast }\left(z\right)=|z-1{|}^{\gamma },\gamma >0$, $L:=\{z:|z|=1\}$ and any $n\in \mathbb{N}$ there exist $c_3=c_3(h^{\ast },p)>0$, $c_3^{\mathrm{\prime }}=c_3^{\mathrm{\prime }}(h^{\ast \ast },p)>0$ such that $\begin{array}{rl}\left(a\right){\Vert T\Vert }_{\mathrm{\infty }}& \ge c_3{n}^{\frac{1}{p}}{\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast },L)},p>1;\\ \left(b\right){\Vert T\Vert }_{\mathrm{\infty }}& \ge c_3^{\mathrm{\prime }}{n}^{\frac{\gamma +1}{p}}{\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast \ast },L)},p>\gamma +1.\end{array}$Really, if $L:=\{z:|z|=1\}$, then, $L\in {\tilde{Q}}_1$. (a) $h^{\ast }\left(z\right)\equiv 1$; (b) $h^{\ast \ast }\left(z\right)=|z-1{|}^{\gamma },\gamma >0$.

Obviously, $\left|T\right(z\left)\right|\le \sum _{j=0}^{n-1}\left|z^j\right|=n,\left|z\right|=1;\left|T\right(1\left)\right|=n.$So, ${\Vert T\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}}=n.$On the other hand, according to [27, p. 236], we have: ${\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast },L)}\approx n^{1-\frac{1}{p}},p>1,$and ${\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast \ast },L)}\approx n^{1-\frac{\gamma +1}{p}},p>\gamma +1.$

Therefore, $\begin{array}{rl}\left(a\right){\Vert T\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}}& =n\approx n^{\frac{1}{p}}{\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast },L)},p>1;\\ \left(b\right){\Vert T\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}}& =n=n\cdot n^{1-\frac{\gamma +1}{p}}\cdot n^{\frac{\gamma +1}{p}-1}\approx n^{\frac{\gamma +1}{p}}{\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast \ast },L)},p>\gamma +1.\text{■}\end{array}$

Acknowledgements

The authors would like to thank the referees for their remarks and useful advice, which helped the authors to significantly improve the work.

Disclosure statement

No potential conflict of interest was reported by the author(s).