On the behaviour of derivative of algebraic polynomials in the regions with cusps

ABSTRACT

In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region.

KEYWORDS:

• Algebraic polynomials
• quasiconformal mapping
• Dini-smooth curve

MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 30A10
• 30C10
• Secondary 41A17

1. Introduction

Let $\mathbb{C}$ be a complex plane and $\overline{\mathbb{C}}:=\mathbb{C}\cup \left\{\mathrm{\infty }\right\};G$ $\subset \mathbb{C}$ ($0\in G)$ be a finite Jordan region, $L:=\mathrm{\partial }G,$ $\mathrm{\Omega }:=\overline{\mathbb{C}}\mathrm{\setminus }\overline{G};$ $\mathrm{\Delta }:=\{w:|w|>1\}$ $($with respect to $\overline{\mathbb{C}}$). Let function $w=\mathrm{\Phi }\left(z\right),\mathrm{\Phi }:$ $\mathrm{\Omega }\to \mathrm{\Delta },$be the univalent and conformal mapping, normalized by $\mathrm{\Phi }\left(\mathrm{\infty }\right)=\mathrm{\infty },\underset{z\to \mathrm{\infty }}{lim}\frac{\mathrm{\Phi }\left(z\right)}{z}>0$, and $\mathrm{\Psi }:={\mathrm{\Phi }}^{-1}.$ For$R>1,$ let us set $L_R:=\{z:|\mathrm{\Phi }\left(z\right)|=R\},$ $G_R:=int{L}_R,$ ${\mathrm{\Omega }}_R:=ext{L}_R$.

The class of all algebraic polynomials $P_n\left(z\right)$ of degree at most $n\in \mathbb{N}$is denoted by ${\mathrm{\wp }}_n.$

Let an arbitrary fixed system of points $\{z_j{\}}_{j=1}^m$, distinct from each other, be given on L. Consider Jacobi weight function $h\left(z\right)$ being defined as (1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) where ${\gamma }_j>-2,$ for all$j=1,2,\dots ,m,$and there exists a constant $c_1\left(G\right)>0$ such that $h_0\left(z\right)\ge c_1\left(G\right)$ for all $z\in G_{R_0}.$

Let $0<p\le \mathrm{\infty }$ and σ be the two-dimensional Lebesgue measure. We introduce (2) $\begin{array}{rl}{\Vert P_n\Vert }_{A_p(h,G)}^p& :={\iint }_{G}h\left(z\right){\left|P_n\right(z\left)\right|}^p\mathrm{d}{\sigma }_z,0<p<\mathrm{\infty },\\ {\Vert P_n\Vert }_{A_{\mathrm{\infty }}(1,G)}& :=\underset{z\in \overline{G}}{max}\left|P_n\right(z\left)\right|,p=\mathrm{\infty },\end{array}$(2) and when L is rectifiable and $h\left(z\right)$ defined as in (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for ${\gamma }_j>-1,$ for all$j=1,2,\dots ,m$: (3) $\begin{array}{r}\begin{array}{rl}{\Vert P_n\Vert }_p^p& :={\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)}^p:={\int }_{L}h\left(z\right){\left|P_n\right(z\left)\right|}^p\left|\mathrm{d}z\right|<\mathrm{\infty },0<p<\mathrm{\infty },\\ {\Vert P_n\Vert }_{\mathrm{\infty }}& :={\Vert P_n\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}(1,L)}:=\underset{z\in L}{max}\left|P_n\right(z\left)\right|;{\mathcal{L}}_p(1,L)=:{\mathcal{L}}_p\left(L\right).\end{array}\end{array}$(3) The well-known classical lemma [1] shows that for any $P_n\left(z\right)\in {\mathrm{\wp }}_n$ (4) $\begin{array}{r}\left|P_n\right(z\left)\right|\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},z\in \mathrm{\Omega },\end{array}$(4) holds. If we take $z\in {\overline{G}}_R,$ then from (4(4) $\begin{array}{r}\left|P_n\right(z\left)\right|\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},z\in \mathrm{\Omega },\end{array}$(4) ) we see that (5) $\begin{array}{r}{\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le R^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)}.\end{array}$(5) Here, we see that $\Vert P_n{\Vert }_{C\left(\overline{G}\right)}$ increases by a constant if the domain G is extended to $G_{1+\frac{1}{n}}.$ Considering the following estimate [2], we see that the same holds for norm (3(3) $\begin{array}{r}\begin{array}{rl}{\Vert P_n\Vert }_p^p& :={\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)}^p:={\int }_{L}h\left(z\right){\left|P_n\right(z\left)\right|}^p\left|\mathrm{d}z\right|<\mathrm{\infty },0<p<\mathrm{\infty },\\ {\Vert P_n\Vert }_{\mathrm{\infty }}& :={\Vert P_n\Vert }_{{\mathcal{L}}_{\mathrm{\infty }}(1,L)}:=\underset{z\in L}{max}\left|P_n\right(z\left)\right|;{\mathcal{L}}_p(1,L)=:{\mathcal{L}}_p\left(L\right).\end{array}\end{array}$(3) ): (6) $\begin{array}{r}{\Vert P_n\Vert }_{{\mathcal{L}}_p\left(L_R\right)}\le R^{n+\frac{1}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p\left(L\right)},p>0.\end{array}$(6) Estimate (6(6) $\begin{array}{r}{\Vert P_n\Vert }_{{\mathcal{L}}_p\left(L_R\right)}\le R^{n+\frac{1}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p\left(L\right)},p>0.\end{array}$(6) ) was generalized in [3, Lemma 2.4] for weight function (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) with ${\gamma }_j>-1,j=1,2,\dots ,m,$ and it was obtained as follows: (7) $\begin{array}{r}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L_R)}\le R^{n+\frac{1+{\gamma }^{\ast }}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)},{\gamma }^{\ast }=max\{0;{\gamma }_j:1\le j\le m\}.\end{array}$(7) In [4], an analogue of estimates (4(4) $\begin{array}{r}\left|P_n\right(z\left)\right|\le {\left|\mathrm{\Phi }\right(z\left)\right|}^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)},z\in \mathrm{\Omega },\end{array}$(4) ) and (7(7) $\begin{array}{r}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L_R)}\le R^{n+\frac{1+{\gamma }^{\ast }}{p}}{\Vert P_n\Vert }_{{\mathcal{L}}_p(h,L)},{\gamma }^{\ast }=max\{0;{\gamma }_j:1\le j\le m\}.\end{array}$(7) ) in the $A_p(h,G)$-norm when $L=\mathrm{\partial }G$ is a quasiconformal (see: Definition 3.1) with a same weight function (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) is given as follows: $\begin{array}{r}{\Vert P_n\Vert }_{{}_{A_p(h,G_R)}}\le c_1{R}^{{\ast }^{n+\frac{1}{p}}}{\Vert P_n\Vert }_{{}_{A_p(h,G)}},p>0,\end{array}$ where $R^{\ast }:=1+c_2(R-1)$ and $c_2>0$, $c_1:=c_1(G,p,c_2)>0$ are constants independent of n, R.

Further, for arbitrary $P_n\in {\mathrm{\wp }}_n,R_1=1+\frac{1}{n},$ and Jordan region G, in [5, Theorem1.1], it was obtained that $\begin{array}{r}{\Vert P_n\Vert }_{{}_{A_p\left(G_R\right)}}\le c{R}^{{}^{n+\frac{2}{p}}}{\Vert P_n\Vert }_{{}_{A_p\left(G_{R_1}\right)}},p>0,\end{array}$ is true for arbitrary $R>R_1=1+\frac{1}{n}$, where $c=\left(\frac{2}{e^p-1}{)}^{\frac{1}{p}}\right[1+O\left(\frac{1}{n}\right)],n\to \mathrm{\infty }.$

Stylianopoulos [6] showed that if a curve L is quasiconformal and rectifiable, then the following is true $\begin{array}{r}\left|P_n\right(z\left)\right|\le c\left(L\right)\frac{\sqrt{n}}{d(z,L)}{\Vert P_n\Vert }_{A_2\left(G\right)}{\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1},z\in \mathrm{\Omega },\end{array}$ for arbitrary $P_n\in {\mathrm{\wp }}_n,$ where $c\left(L\right)>0$is a constant depended only on$L$and $d(z,L):=$ $inf\left\{\right|\zeta -z|:\zeta \in L\}.$

In this paper, we continue the study of the problem on pointwise estimates of the derivatives $\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|$ in unbounded regions and estimate the following type: (8) $\begin{array}{r}\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|\le {\eta }_n(L,h,p,z){\Vert P_n\Vert }_p,z\in \mathrm{\Omega },\end{array}$(8) where ${\eta }_n(L,h,p,z)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$.

Results, analogously to (8(8) $\begin{array}{r}\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|\le {\eta }_n(L,h,p,z){\Vert P_n\Vert }_p,z\in \mathrm{\Omega },\end{array}$(8) ) for arbitrary $P_n\in {\mathrm{\wp }}_n$ a different weight function$h$, an unbounded region and some norms were obtained in [3,6–19; 20, p.418–428; 21,22] and the others.

To get an estimate for the $\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|$ on the $\mathbb{C}$, we will need an estimate for the $\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|$ in the bounded region $\overline{G}$. To do this, we will use estimates Bernstein–Markov–Nikolsky for $\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|,$ $z\in \overline{G}$ as follows: (9) $\begin{array}{r}\Vert P_n^{{}^{\mathrm{\prime }}}{\Vert }_{\mathrm{\infty }}\le {\lambda }_n(L,h,p)\Vert P_n{\Vert }_p,\end{array}$(9) where ${\lambda }_n:={\lambda }_n(G,h,p)>0,$ ${\lambda }_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }.$

Estimates of type (9(9) $\begin{array}{r}\Vert P_n^{{}^{\mathrm{\prime }}}{\Vert }_{\mathrm{\infty }}\le {\lambda }_n(L,h,p)\Vert P_n{\Vert }_p,\end{array}$(9) ) for the arbitrary $P_n\in {\mathrm{\wp }}_n$ were studied in [4,15,23–31;20, p. 418–428;21,32;33, Sect. 5.3;34, p.122–133;35,36] and references therein.

Therefore, combining estimates (8(8) $\begin{array}{r}\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|\le {\eta }_n(L,h,p,z){\Vert P_n\Vert }_p,z\in \mathrm{\Omega },\end{array}$(8) ) and (9(9) $\begin{array}{r}\Vert P_n^{{}^{\mathrm{\prime }}}{\Vert }_{\mathrm{\infty }}\le {\lambda }_n(L,h,p)\Vert P_n{\Vert }_p,\end{array}$(9) ), we obtain (10) $\begin{array}{r}\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|\le c(L,p{\Vert P_n\Vert }_p\{\begin{array}{ll}{\lambda }_n(L,h,p),& z\in {\overline{G}}_R,R>1,\\ {\eta }_n(L,h,p,d(z,L\left){\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1}\right),& z\in {\mathrm{\Omega }}_R,\end{array}\end{array}$(10) where $c(L,p)>0$ is a constant independent of n, h, $P_n,$ and ${\lambda }_n(L,h,p)\to \mathrm{\infty }$, ${\eta }_n(.)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$.

In this work, we study problem (10(10) $\begin{array}{r}\left|P_n^{{}^{\mathrm{\prime }}}\right(z\left)\right|\le c(L,p{\Vert P_n\Vert }_p\{\begin{array}{ll}{\lambda }_n(L,h,p),& z\in {\overline{G}}_R,R>1,\\ {\eta }_n(L,h,p,d(z,L\left){\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1}\right),& z\in {\mathrm{\Omega }}_R,\end{array}\end{array}$(10) ) for regions bounded by the piecewise Dini-smooth curve having exterior and interior zero angles.

Let us give the necessary definitions and notations.

Let S be a rectifiable Jordan curve (or arc) and $z=z\left(s\right),s\in [0,|S\left|\right],\left|S\right|:=mesS$, its natural representation.

Definition 1.1

[37, p.48;38, p.32]

A Jordan curve (arc) $S$ is called Dini-smooth, if it has a parametrization $z=z\left(s\right),0\le s\le \left|S\right|,$such that $z^{{}^{\mathrm{\prime }}}\left(s\right)\ne 0,0\le s\le \left|S\right|$and $\left|z^{{}^{\mathrm{\prime }}}\right(s_2)-z^{{}^{\mathrm{\prime }}}(s_1\left)\right|<g(s_2-s_1),$ $s_1<s_2,$where g is an increasing function for which $\begin{array}{r}{\int }_0^1\frac{g\left(x\right)}{x}\mathrm{d}x<\mathrm{\infty }.\end{array}$

Now, we will define a new class of regions with a piecewise Dini-smooth boundary which has at the boundary points corners, and interior and exterior cusps simultaneously.

For sufficiently small ${ϵ}_1>0$ and for each $j=1,2,$ we define $f_j:[0,{ϵ}_1]\to \mathbb{R}$ twice differentiable functions such that $f_j\left(0\right)=0,f_j^{\left(k\right)}\left(x\right)>0,x>0$and $k=0,1,2.$

Throughout this work, $c,c_{0,}{c}_1,c_2,\dots$ indicate positive, and ${ϵ}_0,{ϵ}_1,{ϵ}_2,\dots$ are sufficiently small positive constants, different in different ratios, depending on G and not essential for this case parameter. For any $k\ge 0$ and$m>k,$notations $j=\overline{k,m}$ denote$j=k,k+1,\dots ,m.$

Definition 1.2

[3]

We say that a Jordan region $G\in PDS({\lambda }_i;f_j),$ $0<{\lambda }_i\le 2,i=\overline{1,m_1},f_j=f_j\left(x\right),j=\overline{m_1+1,m},$if $L=\mathrm{\partial }G$ consists of a union of a finite number of Dini-smooth arcs $\{L_j{\}}_{j=0}^m,$connecting at  the points $\{z_j{\}}_{j=0}^m\in L$such that L is locally Dini-smooth at $z_0\in L\mathrm{\setminus }\{z_j{\}}_{j=1}^m$and

1. for every point $z_i\in L,i=\overline{1,m_1},m_1\le m,G$has exterior (with respect to $\overline{G}$) angles ${\lambda }_i\pi ,0<{\lambda }_i\le 2,$at the corner $z_i;$

2. for every point $z_j\in L,j=\overline{m_1+1,m},$ in the local x- and y-axis co-ordinate system with origin at $z_j$, the following is fulfilled:

   	( b1)	$\{z=x+iy:|z|<{ϵ}_1,c_1{f}_j(x)\le y\le c_2{f}_j(x),0\le x\le {ϵ}_1\}\subset \overline{\mathrm{\Omega }},$
   	( b2)	$\{z=x+iy:|z|<{ϵ}_1,|y|\ge {ϵ}_{2}x,0\le x\le {ϵ}_1\}\subset \overline{G},$ for some constants $-\mathrm{\infty }<c_1<c_2<+\mathrm{\infty },0<{ϵ}_i<1,i=1,2.$

For any $z\in \mathbb{C}$ and sufficiently small ${ϵ}_1>0,$  let $\begin{array}{r}K(z,f_j,{ϵ}_1):=\{z=x+iy:0\le x\le {ϵ}_1,c_1{f}_j(x)\le y\le c_2{f}_j(x\left)\right\},\end{array}$ for some constants $-\mathrm{\infty }<c_1<c_2<+\mathrm{\infty },0<{ϵ}_1<1.$Let $f\left(x\right):=f_{j_0}\left(x\right),$ where $j_0,m_1+1\le j_0\le m,$ is chosen such that $K(z_j,f_{j_0},{ϵ}_1)\subseteq K(z_j,f_j,{ϵ}_1)$ for all $m_1+1\le j\le m$ and sufficiently small ${ϵ}_1>0.$

As can be seen from Definition 1.2, each region $G\in PDS({\lambda }_i;f_j)$ may have ${\lambda }_i\pi ,0<{\lambda }_i\le 2,$ exterior angles (when ${\lambda }_i=2$ interior zero angles), $z_i\in L,i=\overline{1,m_1},$ and external zero angles at the $z_j\in L,j=\overline{m_1+1,m},$ where two boundary arcs join under the $f_j\left(x\right)$-tangency. If G hasn't such angles $(m_1=m=0,)$, we will write $G\in DS$; if G has only ${\lambda }_i\pi ,0<{\lambda }_i\le 2,i={\overline{1,m}}_1,$exterior angles (when ${\lambda }_i=2$ interior zero angles) ($m_1=m\ge 1$), we will write $G\in PDS({\lambda }_i;0)$; if G has only exterior zero angles ($m_1=0$ and $m\ge 1$), we will write $G\in PDS(1;f_j).$

Throughout this work, we will assume that the points $\{z_j{\}}_{j=1}^m\in L$ defined in (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) and Definition 1.2 are identical. Without loss of generality, we assume that these points on the curve $L=\mathrm{\partial }G$ are located in the positive direction such that $G$ has  ${\lambda }_j\pi ,0<{\lambda }_j\le 2,$ $j=\overline{1,m_1},$exterior angles (when ${\lambda }_j=2$ interior zero angles (interior cusps)) at the points $\{z_j{\}}_{j=1}^{m_1}$, $m_1\le m,$ and has exterior zero angles (exterior cusps) at the points $\{z_j{\}}_{j=m_1+1}^m$ and $w_j:=\mathrm{\Phi }\left(z_j\right)$.

We introduce some notations. For clarity of results, we will consider the cases when L has two singular points $z_1\in L$ and $z_2\in L$ (i.e. $m_1=1,m=2$). The case $m_1\ge 2,m\ge 3$ can be given similarly. Let us denote ${\tilde{\lambda }}_k:=max\{1;{\lambda }_k\}$, ${\gamma }_k^{\ast }=max\{0;{\gamma }_k\}$, $k=1,2;$ $\hat{\lambda }:=max\{{\tilde{\lambda }}_1;{\tilde{\lambda }}_2\}$ ; ${\hat{\gamma }}^{\ast }:=\{{\gamma }_1^{\ast };{\gamma }_2^{\ast }\},\hat{\gamma }:=\{{\gamma }_1;{\gamma }_2\}$ ; ${\alpha }_{\ast }:=min\{{\alpha }_1,{\alpha }_2\}$. Further, 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 put $U_{\mathrm{\infty }}(L,\delta ):=\bigcup _{\zeta \in L}U(\zeta ,\delta )$-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 $L;\mathrm{\Omega }\left(\delta \right):=\mathrm{\Omega }(L,\delta ):=\mathrm{\Omega }\cap U_N(L,\delta )$, $\hat{\mathrm{\Omega }}:=\mathrm{\Omega }\mathrm{\setminus }\mathrm{\Omega }\left(\delta \right)$; ${\mathrm{\Omega }}_R\left(\delta \right):=\mathrm{\Omega }(L_R,\delta ):={\mathrm{\Omega }}_R\cap U_N(L_R,\delta )$, ${\hat{\mathrm{\Omega }}}_R:={\mathrm{\Omega }}_R\mathrm{\setminus }{\mathrm{\Omega }}_R\left(\delta \right)$.

2. Main results

Theorem 2.1

Let $p>1;G\in PDS({\lambda }_1;c{x}^{1+{\alpha }_2}),$for some $0<{\lambda }_1\le 2,$ ${\alpha }_2>0;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for j = 2. Then, for each $n\in \mathbb{N}$  and arbitrary $P_n\in {\mathrm{\wp }}_n,$ we have   (11) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_1\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\{\begin{array}{cc}{D}_n^{\left(1\right)}+D_n^{\left(2\right)},& z\in {\mathrm{\Omega }}_R\left(\delta \right),\\ E_n,& z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),\end{array}\end{array}$(11) where $c_1=c_1(L,p)>0$ is the constant independent of z and n, c defined as in Definition 1.2 and $\begin{array}{rl}{D}_n^{\left(1\right)}:& =\{\begin{array}{ll}{n}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}{\tilde{\lambda }}_1},& {\gamma }_1\ge {\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{({\gamma }_2+1)}{p(1+{\alpha }_2)}}},& 0<{\gamma }_1<{\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{1}{p}}{\tilde{\lambda }}_1},& -1<{\gamma }_1<0,-1<{\gamma }_2<(1+{\alpha }_2){\tilde{\lambda }}_1-1;\end{array}\\ D_n^{\left(2\right)}& :=\{\begin{array}{ll}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}\right){\tilde{\lambda }}_1},& {\gamma }_1>p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2\ge p-1,\\ n^{{\tilde{\lambda }}_1+({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& p-1<{\gamma }_1\le p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& {\gamma }_1,{\gamma }_2=p-1,\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1;\end{array}\\ E_n& :=\{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1\\ <{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$

From this theorem, two special cases can be distinguished: the region G has only an exterior non-zero (interior zero) angle or an exterior zero angle. We present the following corollaries separately.

Corollary 2.1

Let $p>1;G\in PDS({\lambda }_1;{\lambda }_2),$for some$0<{\lambda }_1,{\lambda }_2\le 2;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for $j=2$. Then, for each $n\in \mathbb{N}$  and arbitrary $P_n\in {\mathrm{\wp }}_n$, we have (12) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_2\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\{\begin{array}{ll}{D}_{n,1}^{\left(1\right)}+D_{n,2}^{\left(1\right)},& z\in {\mathrm{\Omega }}_R\left(\delta \right),\\ E_{n,1}& z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),\end{array}\end{array}$(12) where $c_2=c_2(L,p)>0$ is a constant independent of z and $n,$and  $\begin{array}{rl}{D}_{n,1}^{\left(1\right)}& :=n^{\frac{({\hat{\gamma }}^{\ast }+1)}{p}\hat{\lambda }};D_{n,2}^{\left(1\right)}:=\{\begin{array}{ll}{n}^{{\displaystyle \frac{\hat{\gamma }+1}{p}}\hat{\lambda }},& {\gamma }_1,{\gamma }_2>p-1,\\ n^{\hat{\lambda }}{(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ n^{\hat{\lambda }},& -1<{\gamma }_1,{\gamma }_2<p-1;\end{array}\\ E_{n,1}& :=\{\begin{array}{ll}{n}^{({\displaystyle \frac{\hat{\gamma }+1}{p}}-1)\hat{\lambda }},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$

Corollary 2.2

Let $p>1;G\in PDS(c{x}^{1+{\alpha }_1};c{x}^{1+{\alpha }_2}),$for some${\alpha }_1,{\alpha }_2>0;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for j = 2. Then, for each $n\in \mathbb{N}$  and arbitrary $P_n\in {\mathrm{\wp }}_n$, we have   (13) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_3\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\{\begin{array}{ll}{D}_{n,1}^{\left(2\right)}+D_{n,2}^{\left(2\right)},& z\in {\mathrm{\Omega }}_R\left(\delta \right),\\ E_{n,2},& z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),\end{array}\end{array}$(13) where $c_3=c_3(L,p)>0$ is the constant independent of z and n, c defined as in Definition 1.2 and  $\begin{array}{rl}{D}_{n,1}^{\left(2\right)}& :=n^{\frac{({\hat{\gamma }}_2^{\ast }+1)}{p(1+{\alpha }_{\ast })}};D_{n,2}^{\left(2\right)}:=\{\begin{array}{ll}{n}^{{\displaystyle \frac{\hat{\gamma }+1}{p(1+{\alpha }_{\ast })}}},& {\gamma }_1,{\gamma }_2>p-1,\\ n^{{\displaystyle \frac{1}{1+{\alpha }_{\ast }}}}{(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \begin{array}{c}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ n^{\frac{1}{1+{\alpha }_{\ast }}},& -1<{\gamma }_1,{\gamma }_2<p-1;\end{array}\\ E_{n,2}& :=\{\begin{array}{ll}{n}^{({\displaystyle \frac{\hat{\gamma }+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_{\ast }}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$

Now, we will find an estimate for $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ for $z\in \overline{G}$.

Theorem 2.2

Let $p>1;G\in PDS({\lambda }_1;c{x}^{1+{\alpha }_2}),$for some $0<{\lambda }_1\le 2,{\alpha }_2>0;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for j = 2. Then, for each $n\in \mathbb{N}$  and arbitrary $P_n\in {\mathrm{\wp }}_n,$ we have (14) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_4{\Vert P_n\Vert }_p[F_{n,1}+F_{n,2}],\end{array}$(14) where $c_4=c_4(L,p)>0$is the constant independent of z and n, c defined as in Definition 1.2 and $\begin{array}{r}{F}_{n,1}:=n^{({\displaystyle \frac{({\gamma }_1^{\ast }+1)}{p}}+1){\tilde{\lambda }}_1};F_{n,2}:=\{\begin{array}{ll}{n}^{{\displaystyle \frac{{\gamma }_2^{\ast }+1}{p(1+{\alpha }_2)}}+{\displaystyle \frac{{\alpha }_2}{p(1+{\alpha }_2)}}+{\tilde{\lambda }}_1},& 1<p<2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{{\tilde{\lambda }}_1+1-{\displaystyle \frac{1}{p}}}(\ln n{)}^{1-{\displaystyle \frac{1}{p}}},& p=2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{{\tilde{\lambda }}_1+1-{\displaystyle \frac{1}{p}}},& p>2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}}.\end{array}\end{array}$

Corollary 2.3

Let $p>1;G\in PDS({\lambda }_1;{\lambda }_2),$for some$0<{\lambda }_1,{\lambda }_2\le 2;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for $j=2$. Then, for each $n\in \mathbb{N}$  and arbitrary $P_n\in {\mathrm{\wp }}_n,$ we have (15) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_5{n}^{(\frac{({\hat{\gamma }}^{\ast }+1)}{p}+1)\hat{\lambda }}{\Vert P_n\Vert }_p,\end{array}$(15) where $c_5=c_5(L,p)>0$is the constant independent of z and n.

Corollary 2.4

Let $p>1;G\in PDS(c{x}^{1+{\alpha }_1};c{x}^{1+{\alpha }_2}),$for some${\alpha }_1,{\alpha }_2>0;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for j = 2. Then, for each $n\in \mathbb{N}$ and arbitrary $P_n\in {\mathrm{\wp }}_n,$ we have (16) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_6{\Vert P_n\Vert }_p\{\begin{array}{ll}{n}^{{\displaystyle \frac{{\hat{\gamma }}^{\ast }+1}{p(1+{\alpha }_{\ast })}}+{\displaystyle \frac{{\alpha }^{\ast }}{p(1+{\alpha }_{\ast })}}+1},& 1<p<2+{\displaystyle \frac{{\hat{\gamma }}^{\ast }}{1+{\alpha }_2}},\\ n^{2-{\displaystyle \frac{1}{p}}}(\ln n{)}^{1-{\displaystyle \frac{1}{p}}},& p=2+{\displaystyle \frac{{\gamma }_1^{\ast }}{1+{\alpha }_2}}=2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{2-{\displaystyle \frac{1}{p}}},& p>2+{\displaystyle \frac{{\hat{\gamma }}^{\ast }}{1+{\alpha }_2}},\end{array}\end{array}$(16) where $c_6=c_6(L,p)>0$ is the constant independent of z and n, c defined as in Definition 1.2.

Remark 2.1

Estimates (14(14) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_4{\Vert P_n\Vert }_p[F_{n,1}+F_{n,2}],\end{array}$(14) ) and, consequently, (15(15) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_5{n}^{(\frac{({\hat{\gamma }}^{\ast }+1)}{p}+1)\hat{\lambda }}{\Vert P_n\Vert }_p,\end{array}$(15) ), (16(16) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_6{\Vert P_n\Vert }_p\{\begin{array}{ll}{n}^{{\displaystyle \frac{{\hat{\gamma }}^{\ast }+1}{p(1+{\alpha }_{\ast })}}+{\displaystyle \frac{{\alpha }^{\ast }}{p(1+{\alpha }_{\ast })}}+1},& 1<p<2+{\displaystyle \frac{{\hat{\gamma }}^{\ast }}{1+{\alpha }_2}},\\ n^{2-{\displaystyle \frac{1}{p}}}(\ln n{)}^{1-{\displaystyle \frac{1}{p}}},& p=2+{\displaystyle \frac{{\gamma }_1^{\ast }}{1+{\alpha }_2}}=2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{2-{\displaystyle \frac{1}{p}}},& p>2+{\displaystyle \frac{{\hat{\gamma }}^{\ast }}{1+{\alpha }_2}},\end{array}\end{array}$(16) ) are sharp.

Combining Theorems 2.1 and 2.2, we obtain:

Theorem 2.3

Let $p>1;$ $G\in PDS({\lambda }_1;c{x}^{1+{\alpha }_2}),$ for some $0<{\lambda }_1\le 2$ and${\alpha }_2>0;$ $h\left(z\right)$be defined as in (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ). Then, for each $n\in \mathbb{N}$  and arbitrary $P_n\in {\mathrm{\wp }}_n,$we have $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_7{\Vert P_n\Vert }_p\{\begin{array}{l}{\displaystyle \frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}}\{\begin{array}{ll}{D}_n^{\left(1\right)}+D_n^{\left(2\right)},& z\in {\mathrm{\Omega }}_R\left(\delta \right),\\ E_n,& z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),\end{array}\\ F_{n,1}+F_{n,2},z\in {\overline{G}}_R,\end{array}\end{array}$ where $c_7=c_7(L,p)>0$is the constant independent of z and n, c defined as in Definition 1.2; $D_n^{\left(k\right)},E_n;$ $F_{n,k},$ $k=1,2,$ are defined as in (11(11) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|\le c_1\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\{\begin{array}{cc}{D}_n^{\left(1\right)}+D_n^{\left(2\right)},& z\in {\mathrm{\Omega }}_R\left(\delta \right),\\ E_n,& z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),\end{array}\end{array}$(11) ) and (14(14) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_4{\Vert P_n\Vert }_p[F_{n,1}+F_{n,2}],\end{array}$(14) ), respectively.

Analogously to Theorem 2.3, from Corollaries 2.1–2.3 and 2.2–2.4, we can write estimates in the whole plane for the cases $G\in PDS({\lambda }_1;{\lambda }_2),0<{\lambda }_1,{\lambda }_2\le 2,$ and $G\in PDS(c{x}^{1+{\alpha }_1};c{x}^{1+{\alpha }_2}),{\alpha }_1,{\alpha }_2>0.$

3. Some auxiliary results

For the nonnegative a>0 and b>0, we will use the notations ‘$a\preccurlyeq b$’, if $a\le cb$ and ‘$a\approx b$ ’, if $c_{1}a\le b\le c_{2}a,$ for some constants c, $c_1,$ $c_2$ (independent of a and b), respectively. In the literature, there are definitions of quasiconformal curves (see, e.g. [37, p.286;39;40, p.97;41]) in various versions. Let us give one of them in the variant we need [41]:

Definition 3.1

The Jordan curve (or arc) L is called K-quasiconformal  ($K\ge 1$), if there is a K-quasiconformal mapping f  of the region $D\supset L$ such that $f\left(L\right)$ is a circle (or line segment).

Let $F\left(L\right)$ denote the set of all sense preserving plane homeomorphisms f of the region $D\supset L$ such that $f\left(L\right)$ is a line segment (or circle), and let $\begin{array}{r}{K}_L:=inf\left\{K\right(f):f\in F(L\left)\right\},\end{array}$ where $K\left(f\right)$ is the maximal dilatation of such mapping f. Then L is a quasiconformal, if $K_L<\mathrm{\infty },$ and L is a K-quasiconformal, if $K_L\le K.$

It is well known that quasiconformal curves may not be rectifiable [42]. The piecewise Dini-smooth curve (or arc) (without cusps) is quasiconformal [39, p.100].

Lemma 3.1

[43,44]

Let L be a K-quasiconformal curve, $z_1\in L,$ $z_2,z_3\in \mathrm{\Omega }\cap \{z:|z-z_1|⪯d(z_1,L_{R_0^{-1}}\left)\right\}$; $w_j=\mathrm{\Phi }\left(z_j\right),(z_2,z_3\in G\cap \{z:|z-z_1|⪯d(z_1,L_{R_0})\};w_j=\phi (z_j\left)\right),$ $j=1,2,3.$ Then,

(a)	$|z_1-z_2|⪯|z_1-z_3|$ 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;
(b)	if $|z_1-z_2|⪯|z_1-z_3|,$ then $\begin{array}{r}{\left|\frac{w_1-w_3}{w_1-w_2}\right|}^{K^2}⪯\left|\frac{z_1-z_3}{z_1-z_2}\right|⪯{\left|\frac{w_1-w_3}{w_1-w_2}\right|}^{K^{-2}},\end{array}$

where $R_0=R_0\left(G\right)>1$ is constant.

Recall that for $0<{\delta }_j<{\delta }_0:=\frac{1}{4}min\left\{\right|z_i-z_j|:i,j=1,2,\dots ,m,i\ne j\}$, we put $\mathrm{\Omega }(z_j,{\delta }_j):=\mathrm{\Omega }\cap \{z:|z-z_j|\le {\delta }_j\};$ $\delta :=\underset{1\le j\le m}{min}{\delta }_j$, $\mathrm{\Omega }\left(\delta \right):=\bigcup _{j=1}^m\mathrm{\Omega }(z_j,\delta ),$ $\hat{\mathrm{\Omega }}:=\mathrm{\Omega }\mathrm{\setminus }\mathrm{\Omega }\left(\delta \right).$ Additionally, let ${\mathrm{\Delta }}_j:=\mathrm{\Phi }\left(\mathrm{\Omega }\right(z_j,\delta \left)\right),$ $\mathrm{\Delta }\left(\delta \right):=\bigcup _{j=1}^m\mathrm{\Phi }\left(\mathrm{\Omega }\right(z_j,\delta \left)\right),$ $\hat{\mathrm{\Delta }}\left(\delta \right):=\mathrm{\Delta }\mathrm{\setminus }\mathrm{\Delta }\left(\delta \right).$ Let $w_j:=\mathrm{\Phi }\left(z_j\right)$ and for ${\phi }_j:=\arg w_j,j=1,2,\dots ,m,$ we put ${\mathrm{\Delta }}_j^{{}^{\mathrm{\prime }}}:=\{t=R{e}^{i\theta }:R>1,\frac{{\phi }_{j-1}+{\phi }_j}{2}\le \theta <\frac{{\phi }_j+{\phi }_{j+1}}{2}\},$ where ${\phi }_0\equiv {\phi }_m,$ ${\phi }_1\equiv {\phi }_{m+1};{\mathrm{\Omega }}_j:=\mathrm{\Psi }\left({\mathrm{\Delta }}_j^{{}^{\mathrm{\prime }}}\right),$ $L^j:=L\cap {\overline{\mathrm{\Omega }}}_j,i=1,2,\dots ,m.$ Clearly, $\mathrm{\Omega }=\bigcup _{j=1}^m{\mathrm{\Omega }}_j.L_R^j:=L_R\cap {\overline{\mathrm{\Omega }}}^j.$ $F^i:=\mathrm{\Phi }\left(L^i\right)={\overline{\mathrm{\Delta }}}_i^{{}^{\mathrm{\prime }}}\cap \{\tau :|\tau |=1\},$ $F_R^i:=\mathrm{\Phi }\left(L_R^i\right)={\overline{\mathrm{\Delta }}}_i^{{}^{\mathrm{\prime }}}\cap \{\tau :|\tau |=R\},i=\overline{1,m}.$

We will often use the following estimate for the derivatives of Ψ [45, Th.2.8]: (17) $\begin{array}{r}\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\approx \frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}.\end{array}$(17) According to the two results [37, p.41–58] and [38, p.32–36] and estimate (17(17) $\begin{array}{r}\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(\tau \left)\right|\approx \frac{\mathrm{d}(\mathrm{\Psi }\left(\tau \right),L)}{\left|\tau \right|-1}.\end{array}$(17) ), we have the following:

Lemma 3.2

Let a Jordan region $G\in PDS({\lambda }_j;0),0<{\lambda }_j\le 2,j=\overline{1,m_1}$. Then,

(i)	for any $w\in$ ${\mathrm{\Delta }}_j,$ $\left|\mathrm{\Psi }\right(w)-\mathrm{\Psi }(w_j\left)\right|\approx |w-w_j{|}^{{\lambda }_j},|{\mathrm{\Psi }}^{\mathrm{\prime }}\left(w\right)|\approx |w-w_j{|}^{{\lambda }_j-1};$
(ii)	for any $w\in \overline{\mathrm{\Delta }}\text{}{\mathrm{\Delta }}_j$, $\left|\mathrm{\Psi }\right(w)-\mathrm{\Psi }(w_j\left)\right|\approx |w-w_j|,\left|{\mathrm{\Psi }}^{\mathrm{\prime }}\right(w\left)\right|\approx 1.$

4. Proofs

Proof

Proof of Theorem  2.1

Let $G\in PDS({\lambda }_1;f_2),$ for some $0<{\lambda }_1\le 2$and $f_2\left(x\right)=c{x}^{1+{\alpha }_2},$ ${\alpha }_2>0;$ $h\left(z\right)$be defined in (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ). For $z\in \mathrm{\Omega },$ let us denote:   (18) $\begin{array}{r}{T}_n\left(z\right):=\frac{P_n\left(z\right)}{{\mathrm{\Phi }}^{n+1}\left(z\right)}.\end{array}$(18) Taking the derivative, we obtain $\begin{array}{r}{P}_n^{\mathrm{\prime }}\left(z\right)={\mathrm{\Phi }}^{n+1}\left(z\right)[T_n^{\mathrm{\prime }}\left(z\right)-P_n\left(z\right){\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\mathrm{\prime }}],z\in \mathrm{\Omega }.\end{array}$ Let us write the Cauchy integral representation for unbounded region Ω for $T_n\left(z\right)$ and $(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}{)}^{{}^{\mathrm{\prime }}},$ respectively: $\begin{array}{rl}{T}_n^{\mathrm{\prime }}\left(z\right)& =-\frac{1}{2\pi i}{\int }_L{T}_n\left(\zeta \right)\frac{\mathrm{d}\zeta }{{(\zeta -z)}^2}\\ & =-\frac{1}{2\pi i}{\int }_L\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\frac{\mathrm{d}\zeta }{{(\zeta -z)}^2},z\in {\mathrm{\Omega }}_R,\end{array}$ and $\begin{array}{r}{\left(\frac{1}{{\mathrm{\Phi }}^{n+1}\left(z\right)}\right)}^{\mathrm{\prime }}=-\frac{1}{2\pi i}{\int }_L\frac{1}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\frac{\mathrm{d}\zeta }{{(\zeta -z)}^2},z\in {\mathrm{\Omega }}_R.\end{array}$ Then $\begin{array}{r}{P}_n^{\mathrm{\prime }}\left(z\right)={\mathrm{\Phi }}^{n+1}\left(z\right)[-\frac{1}{2\pi i}{\int }_L\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\frac{\mathrm{d}\zeta }{{(\zeta -z)}^2}+\frac{P_n\left(z\right)}{2\pi i}{\int }_L\frac{1}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\frac{\mathrm{d}\zeta }{{(\zeta -z)}^2}],\end{array}$ and therefore, (19) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& \le \frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{2\pi }\\ & \times [{\int }_L\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{1}{d(z,L)}{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}],\end{array}$(19) since $\left|\mathrm{\Phi }\right(\zeta \left)\right|=1,$for $\zeta \in L.$

Denote by (20) $\begin{array}{r}{A}_n\left(z\right):={\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|};B_n\left(z\right):={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2},\end{array}$(20) and estimate these integrals separately.

To estimate $A_n\left(z\right),$ we first multiply and divide the integrand by $h^{\frac{1}{p}}\left(\zeta \right)$, then apply Hölder's inequality: (21) $\begin{array}{r}{A}_n\left(z\right)\le {\Vert P_n\Vert }_p{\left({\int }_L\frac{\left|\mathrm{d}\zeta \right|}{h^{\frac{q}{p}}\left(\zeta \right){|\zeta -z|}^q}\right)}^{\frac{1}{q}},\frac{1}{p}+\frac{1}{q}=1.\end{array}$(21) Denote by $J_n\left(z\right)$  the last integral, we obtain (22) $\begin{array}{rl}{\left[J_n\right(z\left)\right]}^q& :={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{h^{\frac{q}{p}}\left(\zeta \right){|\zeta -z|}^q}\\ & ⪯{\int }_{L^1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{\frac{q{\gamma }_1}{p}}{|\zeta -z|}^q}+{\int }_{L^2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{\frac{q{\gamma }_2}{p}}{|\zeta -z|}^q}\\ & =:J_{n,1}^1\left(z\right)+J_{n,2}^2\left(z\right).\end{array}$(22) For ease of calculation, we assume that $z_1=-1,$ $z_2=1;$ $(-1,1)\subset G$and let the local co-ordinate axis in Definition 1.2 be parallel to x-axis and $y$-axis in the co-ordinate system; $L=L^{+}\cup L^{-},$ where $L^{+}:=\{z\in L:\mathrm{Im}z\ge 0\},$ $L^{-}:=\{z\in L:\mathrm{Im}z<0\};$let $w^{\pm }:=\{w=e^{i\theta }:\theta =\frac{{\phi }_1\pm {\phi }_2}{2}\},$ $z^{\pm }\in \mathrm{\Psi }\left(w^{\pm }\right),$ $l_i^{\pm }(z_i,z^{\pm })$ denote the arcs, connected the points $z_i$ which $z^{\pm },$ respectively; $\left|l_i^{\pm }\right|:=mes{l}_i^{\pm }(z_i,z^{\pm }),$ $i=1,2.$ As a point, $z_0\in L^{+}$ can be taken as an arbitrary fixed point . Then, from (21(21) $\begin{array}{r}{A}_n\left(z\right)\le {\Vert P_n\Vert }_p{\left({\int }_L\frac{\left|\mathrm{d}\zeta \right|}{h^{\frac{q}{p}}\left(\zeta \right){|\zeta -z|}^q}\right)}^{\frac{1}{q}},\frac{1}{p}+\frac{1}{q}=1.\end{array}$(21) ) and (22(22) $\begin{array}{rl}{\left[J_n\right(z\left)\right]}^q& :={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{h^{\frac{q}{p}}\left(\zeta \right){|\zeta -z|}^q}\\ & ⪯{\int }_{L^1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{\frac{q{\gamma }_1}{p}}{|\zeta -z|}^q}+{\int }_{L^2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{\frac{q{\gamma }_2}{p}}{|\zeta -z|}^q}\\ & =:J_{n,1}^1\left(z\right)+J_{n,2}^2\left(z\right).\end{array}$(22) ), we have (23) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p\{{\left[J_{n,1}^1\right(z\left)\right]}^{\frac{1}{q}}+{\left[J_{n,2}^2\right(z\left)\right]}^{\frac{1}{q}}\},\end{array}$(23) where $\begin{array}{r}{J}_{n,1}^1\left(z\right):={\int }_{L^1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_1(q-1)}{|\zeta -z|}^q};J_{n,2}^2\left(z\right):={\int }_{L^2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)}{|\zeta -z|}^q}.\end{array}$ For estimation of the last integrals, we define the following notations: $R=1+\frac{1}{n};d_{i,R}:=d(z_i,L_R);$

$E_1^{1,\pm }:=\{\zeta \in L^1:|\zeta -z_1|<c_1{d}_{1,R}\}$, $E_2^{1,\pm }:=\{\zeta \in L^1:c_1{d}_{1,R}\le |\zeta -z_1|\le |l_1^{\pm }\left|\right\}$, $E_1^{2,\pm }:=\{\zeta \in L^2:|\zeta -z_2|<c_2{d}_{2,R}\}$, $E_2^{2,\pm }:=\{\zeta \in L^2:c_2{d}_{2,R}\le |\zeta -z_2|\le |l_2^{\pm }\left|\right\}$; $I_{n,k}^{i,\pm }\left(z\right):=I_{n,k}^i\left(E_k^{i,\pm }\right):={\int }_{E_k^{i,\pm }}\frac{\left|\mathrm{d}\zeta \right|}{|\zeta -z_i{|}^{{\gamma }_i(q-1)}|\zeta -z{|}^q};i,k=1,2.$

Given these designations, from (23(23) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p\{{\left[J_{n,1}^1\right(z\left)\right]}^{\frac{1}{q}}+{\left[J_{n,2}^2\right(z\left)\right]}^{\frac{1}{q}}\},\end{array}$(23) ), we have (24) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p\sum _{i=1}^2{\left[I_{n,1}^{i,\pm }\right(z)+I_{n,2}^{i,\pm }(z\left)\right]}^{\frac{1}{q}},i=1,2.\end{array}$(24) According to (19(19) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& \le \frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{2\pi }\\ & \times [{\int }_L\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{1}{d(z,L)}{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}],\end{array}$(19) ) and (20(20) $\begin{array}{r}{A}_n\left(z\right):={\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|};B_n\left(z\right):={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2},\end{array}$(20) ), it suffices to estimate the integrals $I_{n,k}^{i,\pm }$for each i = 1, 2 and $k=1,2.$

Taking into account the positive and negative values of ${\gamma }_i$, we will make separate estimates for $J_{n,2}^1.$

1. Let $z\in {\mathrm{\Omega }}_R\left(\delta \right).$ Denote by $\begin{array}{rl}{\left(E_k^{i,\pm }\right)}_1& :=\{\zeta \in E_k^{i,\pm }:|\zeta -z_i|<|\zeta -z|\},{\left(E_k^{i,\pm }\right)}_2:=E_k^{i,\pm }\mathrm{\setminus }{\left(E_k^{i,\pm }\right)}_1;\\ {\left[I_{n,k}^{i,\pm }\right(z\left)\right]}_1& :=I_{n,k}^i\left[{\left(E_k^{i,\pm }\right)}_1\right]:={\int }_{{\left(E_k^{i,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_i|}^{{\gamma }_i(q-1)+}{}^q};\\ {\left[I_{n,k}^{i,\pm }\right(z\left)\right]}_2& :=I_{n,k}^i\left[{\left(E_k^{i,\pm }\right)}_2\right]:={\int }_{{\left(E_k^{i,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_i(q-1)+}{}^q}.\end{array}$

   (1.1)

   Let ${\gamma }_1\ge 0$and ${\gamma }_2\ge 0.$ In this case, we obtain (25) $\begin{array}{rl}{\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_1& ={\int }_{{\left(E_1^{1,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_i(q-1)+}{}^q}\\ & ⪯{\int }_0^{c_1{d}_{1,R}}\frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)+q}}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_2& ={\int }_{{\left(E_1^{1,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_i(q-1)+}{}^q}\\ & ⪯d_{1,R}^{-\left[\right({\gamma }_1+1)+q]}\cdot mes\left(E_1^{1,\pm }\right)⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ I_{n,1}^{1,\pm }\left(z\right)& ={\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_1+{\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_2⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_2^{1,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_1(q-1)+}{}^q}⪯{\int }_{c_1{d}_{1,R}}^{\left|l_1^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)+q}}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_2^{1,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_1(q-1)+}{}^q}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ I_{n,2}^{1,\pm }\left(z\right)& ={\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_1+{\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_2⪯d_{1,R}^{-({\gamma }_1+1)(q-1)}.\end{array}$(25) Similar estimates for $J_{n,2}^2\left(z\right)$ in the neighbourhood of the point $z_2$ is given as (26) $\begin{array}{rl}{\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_1^{2,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)+q}}\\ & ⪯{\int }_0^{c_2{d}_{2,R}}\frac{ds}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_1^{2,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_2(q-1)+q}}\\ & ⪯d_{2,R}^{-\left[\right({\gamma }_2+1)+q]}\cdot mes\left(E_1^{2,\pm }\right)⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ I_{n,1}^{2,\pm }\left(z\right)& ={\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_1+{\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_2⪯d_{2,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_2^{2,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)+q}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ {\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_2^{2,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_2(q-1)+q}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ I_{n,2}^{2,\pm }\left(z\right)& ={\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_1+{\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_2⪯d_{2,R}^{-({\gamma }_1+1)(q-1)}.\end{array}$(26) Let ${\gamma }_1<0$and ${\gamma }_2<0.$Then, analogously to  (25(25) $\begin{array}{rl}{\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_1& ={\int }_{{\left(E_1^{1,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_i(q-1)+}{}^q}\\ & ⪯{\int }_0^{c_1{d}_{1,R}}\frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)+q}}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_2& ={\int }_{{\left(E_1^{1,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_i(q-1)+}{}^q}\\ & ⪯d_{1,R}^{-\left[\right({\gamma }_1+1)+q]}\cdot mes\left(E_1^{1,\pm }\right)⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ I_{n,1}^{1,\pm }\left(z\right)& ={\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_1+{\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_2⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_2^{1,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_1(q-1)+}{}^q}⪯{\int }_{c_1{d}_{1,R}}^{\left|l_1^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)+q}}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_2^{1,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_1(q-1)+}{}^q}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ I_{n,2}^{1,\pm }\left(z\right)& ={\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_1+{\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_2⪯d_{1,R}^{-({\gamma }_1+1)(q-1)}.\end{array}$(25) ) and (26(26) $\begin{array}{rl}{\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_1^{2,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)+q}}\\ & ⪯{\int }_0^{c_2{d}_{2,R}}\frac{ds}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_1^{2,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_2(q-1)+q}}\\ & ⪯d_{2,R}^{-\left[\right({\gamma }_2+1)+q]}\cdot mes\left(E_1^{2,\pm }\right)⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ I_{n,1}^{2,\pm }\left(z\right)& ={\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_1+{\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_2⪯d_{2,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_2^{2,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)+q}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ {\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_2^{2,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_2(q-1)+q}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ I_{n,2}^{2,\pm }\left(z\right)& ={\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_1+{\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_2⪯d_{2,R}^{-({\gamma }_1+1)(q-1)}.\end{array}$(26) ), we obtain (27) $\begin{array}{r}\begin{array}{rl}{I}_{n,1}^{1,\pm }\left(z\right)& ={\int }_{E_1^{1,\pm }}\frac{{|\zeta -z_1|}^{-{\gamma }_1(q-1)}\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^q}\\ & ⪯d_{1,R}^{-{\gamma }_1(q-1)-q}mes{E}_1^{1,\pm }⪯d_{1,R}^{(-{\gamma }_1-1)(q-1)};\\ I_{n,2}^{1,\pm }\left(z\right)& ⪯{\int }_{E_2^{1,\pm }}\frac{{|\zeta -z_1|}^{-{\gamma }_1(q-1)}\left|\mathrm{d}\zeta \right|}{|\zeta -z|{}^q}⪯{\int }_{c_1{d}_{1,R}}^{\left|l_1^{\pm }\right|}\frac{\mathrm{d}s}{s^q}⪯d_{1,R}^{-(q-1)};\end{array}\end{array}$(27) and (28) $\begin{array}{r}\begin{array}{rl}{I}_{n,1}^{2,\pm }\left(z\right)& ⪯{\int }_{E_1^{2,\pm }}\frac{{|\zeta -z_2|}^{-{\gamma }_2(q-1)}}{|\zeta -z|{}^q}\left|\mathrm{d}\zeta \right|\\ & ⪯d_{2,R}^{(-{\gamma }_2)(q-1)}mes{E}_1^{2,\pm }⪯d_{2,R}^{(-{\gamma }_2-1)(q-1)};\\ I_{n,2}^{2,\pm }\left(z\right)& ⪯{\int }_{E_2^{2,\pm }}\frac{{|\zeta -z_2|}^{-{\gamma }_2(q-1)}}{|\zeta -z|{}^q}\left|\mathrm{d}\zeta \right|⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^q}⪯d_{2,R}^{-(q-1)}.\end{array}\end{array}$(28) Therefore, in this case, from (24(24) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p\sum _{i=1}^2{\left[I_{n,1}^{i,\pm }\right(z)+I_{n,2}^{i,\pm }(z\left)\right]}^{\frac{1}{q}},i=1,2.\end{array}$(24) ) to (28(28) $\begin{array}{r}\begin{array}{rl}{I}_{n,1}^{2,\pm }\left(z\right)& ⪯{\int }_{E_1^{2,\pm }}\frac{{|\zeta -z_2|}^{-{\gamma }_2(q-1)}}{|\zeta -z|{}^q}\left|\mathrm{d}\zeta \right|\\ & ⪯d_{2,R}^{(-{\gamma }_2)(q-1)}mes{E}_1^{2,\pm }⪯d_{2,R}^{(-{\gamma }_2-1)(q-1)};\\ I_{n,2}^{2,\pm }\left(z\right)& ⪯{\int }_{E_2^{2,\pm }}\frac{{|\zeta -z_2|}^{-{\gamma }_2(q-1)}}{|\zeta -z|{}^q}\left|\mathrm{d}\zeta \right|⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^q}⪯d_{2,R}^{-(q-1)}.\end{array}\end{array}$(28) ), we obtain (29) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p[d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}}],\end{array}$(29) where ${\gamma }_i^{\ast }:=max\{0;{\gamma }_i,i=1,2\}.$

Now, let us estimate $\begin{array}{r}{B}_n\left(z\right):={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}.\end{array}$ By notation from (23(23) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p\{{\left[J_{n,1}^1\right(z\left)\right]}^{\frac{1}{q}}+{\left[J_{n,2}^2\right(z\left)\right]}^{\frac{1}{q}}\},\end{array}$(23) ), $L=L^{\pm }=E_1^{1,\pm }\cup E_2^{1,\pm }\cup E_1^{2,\pm }\cup E_2^{2,\pm }$ and so (30) $\begin{array}{r}{B}_n\left(z\right)=\sum _{i,k=1}^2{\int }_{E_k^{i,+}\cup E_k^{i,-}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}=:M\left(E_k^{i,+}\right)+M\left(E_k^{i,-}\right).\end{array}$(30) The estimation of the integrals $M\left(E_k^{i,+}\right)$ and $M\left(E_k^{i,-}\right),$ $i,k=1,2,$ is similar, then we will estimate only $M\left(E_k^{i,+}\right)$: $\begin{array}{rl}M\left(E_1^{1,+}\right)& ={\int }_{E_1^{1,+}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}={\int }_{|z_1-z|}^{c_1{d}_{1,R}}\frac{\mathrm{d}s}{s^2}⪯\frac{1}{d_{1,R}};\\ M\left(E_2^{1,+}\right)& ={\int }_{E_2^{1,+}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}={\int }_{d_{1,R}}^{\left|l_1^{\pm }\right|}\frac{\mathrm{d}s}{s^2}⪯\frac{1}{d_{1,R}};\\ M\left(E_1^{2,+}\right)& ={\int }_{E_1^{2,+}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}={\int }_0^{|z_2-z_2^{+}|}\frac{\mathrm{d}s}{s^2}⪯{\int }_0^{c_1{d}_{2,R}}\frac{ds}{s^2}⪯\frac{1}{d_{2,R}};\\ M\left(E_2^{2,+}\right)& ={\int }_{E_2^{2,+}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}={\int }_{|z_2-z_2^{+}|}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^2}⪯{\int }_{c_1{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^2}⪯\frac{1}{d_{2,R}}.\end{array}$ Therefore, (31) $\begin{array}{r}{B}_n\left(z\right)⪯\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}}.\end{array}$(31) Comparing (19(19) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& \le \frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{2\pi }\\ & \times [{\int }_L\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{1}{d(z,L)}{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}],\end{array}$(19) ), (20(20) $\begin{array}{r}{A}_n\left(z\right):={\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|};B_n\left(z\right):={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2},\end{array}$(20) ), (29(29) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p[d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}}],\end{array}$(29) ) and (31(31) $\begin{array}{r}{B}_n\left(z\right)⪯\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}}.\end{array}$(31) ), we have (32) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{1}{d(z,L)}{\Vert P_n\Vert }_p(d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}})+\left|P_n\left(z\right)\right|(\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}})].\end{array}$(32) According to [3, Th.1.1], we have (33) $\begin{array}{r}\left|P_n\left(z\right)\right|\le c\frac{B_{n,1}}{d(z,L)}{\Vert P_n\Vert }_p{\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1},\end{array}$(33) where $c=c(G,p,{\gamma }_i)>0,i=\overline{1,p},$ constant independent of $n$and$z,$ and (34) $\begin{array}{r}{B}_{n,1}:=\{\begin{array}{ll}{d}_{1,R}^{-({\displaystyle \frac{{\gamma }_1+1}{p}}-1)}+d_{2,R}^{-(\frac{{\gamma }_2+1}{p}-1)},& {\gamma }_1,{\gamma }_2>p-1,\\ \ln {\displaystyle \frac{1}{d_{1,R}}}+\ln {\displaystyle \frac{1}{d_{2,R}}},& {\gamma }_1={\gamma }_2=p-1,\\ 1,& {\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(34) Substituting (33(33) $\begin{array}{r}\left|P_n\left(z\right)\right|\le c\frac{B_{n,1}}{d(z,L)}{\Vert P_n\Vert }_p{\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1},\end{array}$(33) ) into (32(32) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{1}{d(z,L)}{\Vert P_n\Vert }_p(d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}})+\left|P_n\left(z\right)\right|(\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}})].\end{array}$(32) ), we obtain (35) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}})+B_{n,1}(\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}})].\end{array}$(35)

According to Lemma 3.2, for point $z_1,$ we get (36) $\begin{array}{r}{d}_{1,R}⪰n^{-{\tilde{\lambda }}_1}.\end{array}$(36) For estimate $d_{2,R},$we put $z_R\in L_R:$ $d_{2,R}=|z_2-z_R|{\zeta }^{\pm }\in L^{\pm }:d(z_R,L^2\cap L^{\pm }):=d(z_R,L^{+});$ $z_2^{\pm }$ $:=\zeta \in L^2:|\zeta -z_2|=c_2{d}_{2,R}$. Then, from Lemma 3.1, we obtain (37) $\begin{array}{r}{d}_R^{\pm }:=d(z_R,L^2\cap L^{\pm })\approx |z_R-z_2^{\pm }|\approx d_{2,R}^{1+{\alpha }_2}.\end{array}$(37) In this, $d_{2,R}=(d_R^{\pm }{)}^{\frac{1}{1+{\alpha }_2}}.$ On the other hand, using Lemma 3.2 and [46, Corollary 2], we obtain $d_R^{\pm }⪰n^{-1}.$ Therefore, (38) $\begin{array}{r}{d}_{2,R}⪰n^{-\frac{1}{1+{\alpha }_2}}.\end{array}$(38) Applying (36(36) $\begin{array}{r}{d}_{1,R}⪰n^{-{\tilde{\lambda }}_1}.\end{array}$(36) ) and (38(38) $\begin{array}{r}{d}_{2,R}⪰n^{-\frac{1}{1+{\alpha }_2}}.\end{array}$(38) ), for $B_{n,1}$ in (33(33) $\begin{array}{r}\left|P_n\left(z\right)\right|\le c\frac{B_{n,1}}{d(z,L)}{\Vert P_n\Vert }_p{\left|\mathrm{\Phi }\right(z\left)\right|}^{n+1},\end{array}$(33) ), we obtain (39) $\begin{array}{r}{B}_{n,1}\approx \{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(39) From (35(35) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}})+B_{n,1}(\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}})].\end{array}$(35) ) to (39(39) $\begin{array}{r}{B}_{n,1}\approx \{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(39) ), we obtain $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(n^{\frac{({\gamma }_1^{\ast }+1)}{p}{\tilde{\lambda }}_1}+n^{\frac{({\gamma }_2^{\ast }+1)}{p(1+{\alpha }_2)}})+B_{n,1}(n^{{\tilde{\lambda }}_1}+n^{\frac{1}{1+{\alpha }_2}})]\\ & ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\end{array}$ $\begin{array}{rl}& \times \left[\begin{array}{l}\{\begin{array}{ll}{n}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}{\tilde{\lambda }}_1},& {\gamma }_1\ge {\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{({\gamma }_2+1)}{p(1+{\alpha }_2)}}},& 0<{\gamma }_1<{\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{\frac{1}{p}{\tilde{\lambda }}_1},& -1<{\gamma }_1<0,-1<{\gamma }_2<(1+{\alpha }_2){\tilde{\lambda }}_1-1,\end{array}\\ +n^{{\tilde{\lambda }}_1}{B}_{n,1},\end{array}\right]\\ & ⪯\frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\end{array}$ $\begin{array}{rl}& \times \left[\begin{array}{l}\{\begin{array}{ll}{n}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}{\tilde{\lambda }}_1},& {\gamma }_1\ge {\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{({\gamma }_2+1)}{p(1+{\alpha }_2)}}},& 0<{\gamma }_1<{\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{\frac{1}{p}{\tilde{\lambda }}_1},& -1<{\gamma }_1<0,-1<{\gamma }_2<(1+{\alpha }_2){\tilde{\lambda }}_1-1,\end{array}\\ +\{\begin{array}{l}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}\right){\tilde{\lambda }}_1}+n^{{\tilde{\lambda }}_1+({\displaystyle \frac{{\gamma }_2+1}{p}}-1)\frac{1}{1+{\alpha }_2}},{\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},\{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1\end{array}\\ 1,{\gamma }_1,{\gamma }_2<p-1,\end{array}\end{array}\right]\\ & ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\\ & \times \left[\begin{array}{l}\{\begin{array}{ll}{n}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}{\tilde{\lambda }}_1},& {\gamma }_1\ge {\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{({\gamma }_2+1)}{p(1+{\alpha }_2)}}},& 0<{\gamma }_1<{\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{1}{p}}{\tilde{\lambda }}_1},& -1<{\gamma }_1<0,-1<{\gamma }_2<(1+{\alpha }_2){\tilde{\lambda }}_1-1,\end{array}\\ +\{\begin{array}{ll}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}\right){\tilde{\lambda }}_1},& {\gamma }_1>p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2>p-1,\\ n^{{\tilde{\lambda }}_1+({\displaystyle \frac{{\gamma }_2+1}{p}}-1)\frac{1}{1+{\alpha }_2}}& p-1<{\gamma }_1\le p-1+\frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1},{\gamma }_2>p-1,\\ n^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}\right){\tilde{\lambda }}_1},& {\gamma }_1>p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2=p-1,\\ {(\ln n)}^{1-\frac{1}{p}}& -1<{\gamma }_1\le p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2=p-1,\\ {(\ln n)}^{1-\frac{1}{p}}& {\gamma }_1=p-1,-1<{\gamma }_2<p-1,\\ 1,& -1<{\gamma }_1<p-1,-1<{\gamma }_2<p-1,\end{array}\end{array}\right]\end{array}$ (40) $\begin{array}{rl}& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\\ & \times \left[\begin{array}{l}\{\begin{array}{ll}{n}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}{\tilde{\lambda }}_1},& {\gamma }_1\ge {\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{({\gamma }_2+1)}{p(1+{\alpha }_2)}}},& 0<{\gamma }_1<{\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{\frac{1}{p}{\tilde{\lambda }}_1},& -1<{\gamma }_1<0,-1<{\gamma }_2<(1+{\alpha }_2){\tilde{\lambda }}_1-1,\end{array}\\ +\{\begin{array}{ll}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}\right){\tilde{\lambda }}_1},& {\gamma }_1>p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2\ge p-1,\\ n^{{\tilde{\lambda }}_1+({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& p-1<{\gamma }_1\le p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& -1<{\gamma }_1\le p-1,-1<{\gamma }_2\le p-1,\\ 1,& -1<{\gamma }_1<p-1,-1<{\gamma }_2<p-1.\end{array}\end{array}\right]\end{array}$(40) Therefore, we obtain (41) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[D_n^{\left(1\right)}+D_n^{\left(2\right)}],\end{array}$(41) where $\begin{array}{rl}{D}_n^{\left(1\right)}& :=\{\begin{array}{ll}{n}^{{\displaystyle \frac{({\gamma }_1+1)}{p}}{\tilde{\lambda }}_1},& {\gamma }_1\ge {\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{({\gamma }_2+1)}{p(1+{\alpha }_2)}}},& 0<{\gamma }_1<{\displaystyle \frac{{\gamma }_2+1}{(1+{\alpha }_2){\tilde{\lambda }}_1}}-1,{\gamma }_2\ge (1+{\alpha }_2){\tilde{\lambda }}_1-1,\\ n^{{\displaystyle \frac{1}{p}}{\tilde{\lambda }}_1},& -1<{\gamma }_1<0,-1<{\gamma }_2<(1+{\alpha }_2){\tilde{\lambda }}_1-1;\end{array}\\ D_n^{\left(2\right)}& :=\{\begin{array}{ll}{n}^{\left({\displaystyle \frac{{\gamma }_1+1}{p}}\right){\tilde{\lambda }}_1},& {\gamma }_1>p-1+{\displaystyle \frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1}},{\gamma }_2\ge p-1,\\ n^{{\tilde{\lambda }}_1+({\displaystyle \frac{{\gamma }_2+1}{p}}-1)\frac{1}{1+{\alpha }_2}},& \{\begin{array}{l}p-1<{\gamma }_1\le p-1+\frac{{\gamma }_2+1-p}{(1+{\alpha }_2){\tilde{\lambda }}_1},\\ {\gamma }_2>p-1,\end{array}\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& {\gamma }_1,{\gamma }_2=p-1,\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$

Therefore, we complete the proof for the point $z\in \mathrm{\Omega }\left(\delta \right)$.

(2)

Let $z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right).$ (2.1) Let ${\gamma }_1\ge 0$and ${\gamma }_2\ge 0.$ In this case, we obtain (42) $\begin{array}{r}\begin{array}{rl}{I}_{n,1}^{1,\pm }\left(z\right)& ⪯{\int }_{E_1^{1,\pm }}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_1(q-1)}}⪯{\int }_0^{c_1{d}_{1,R}}\\ \frac{ds}{s^{{\gamma }_1(q-1)}}& ⪯\{\begin{array}{ll}{d}_{1,R}^{1-{\gamma }_1(q-1)},& {\gamma }_1(q-1)>1,\\ \ln {\displaystyle \frac{1}{d_{1,R}}},& {\gamma }_1(q-1)=1,\\ 1,& {\gamma }_1(q-1)<1;\end{array}\\ I_{n,2}^{1,\pm }\left(z\right)& ⪯{\int }_{E_2^{1,\pm }}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_1(q-1)}}⪯{\int }_{c_1{d}_{1,R}}^{\left|l_1^{\pm }\right|}\\ \frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)}}& ⪯\{\begin{array}{ll}{d}_{1,R}^{1-{\gamma }_1(q-1)},& {\gamma }_1(q-1)>1,\\ \ln {\displaystyle \frac{1}{d_{1,R}}},& {\gamma }_1(q-1)=1,\\ 1,& {\gamma }_1(q-1)<1;\end{array}\\ I_{n,1}^{1,\pm }\left(z\right)+I_{n,2}^{1,\pm }\left(z\right)& ⪯\{\begin{array}{cc}{d}_{1,R}^{1-{\gamma }_1(q-1)},& {\gamma }_1(q-1)>1,\\ \ln {\displaystyle \frac{1}{d_{1,R}}},& {\gamma }_1(q-1)=1,\\ 1,& {\gamma }_1(q-1)<1,\end{array}\end{array}\end{array}$(42) and in similarly, (43) $\begin{array}{r}\begin{array}{rl}{I}_{n,1}^{2,\pm }\left(z\right)& ⪯{\int }_{E_1^{2,\pm }}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)}}⪯{\int }_0^{c_2{d}_{2,R}}\\ \frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)}}& ⪯\{\begin{array}{ll}{d}_{2,R}^{1-{\gamma }_2(q-1)},& {\gamma }_2(q-1)>1,\\ \ln {\displaystyle \frac{1}{d_{2,R}}},& {\gamma }_2(q-1)=1,\\ 1,& {\gamma }_2(q-1)<1;\end{array}\\ I_{n,2}^{2,\pm }\left(z\right)& ⪯{\int }_{E_2^{2,\pm }}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\\ \frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)}}& ⪯\{\begin{array}{ll}{d}_{2,R}^{1-{\gamma }_2(q-1)},& {\gamma }_2(q-1)>1,\\ \ln {\displaystyle \frac{1}{d_{2,R}}},& {\gamma }_2(q-1)=1,\\ 1,& {\gamma }_2(q-1)<1;\end{array}\\ I_{n,1}^2\left(z\right)+I_{n,2}^{2,\pm }\left(z\right)& ⪯\{\begin{array}{ll}{d}_{2,R}^{1-{\gamma }_2(q-1)},& {\gamma }_2(q-1)>1,\\ \ln {\displaystyle \frac{1}{d_{2,R}}},& {\gamma }_2(q-1)=1,\\ 1,& {\gamma }_2(q-1)<1.\end{array}\end{array}\end{array}$(43) Let ${\gamma }_1<0$ and ${\gamma }_2<0.$ Then, analogous to estimates (25(25) $\begin{array}{rl}{\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_1& ={\int }_{{\left(E_1^{1,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_i(q-1)+}{}^q}\\ & ⪯{\int }_0^{c_1{d}_{1,R}}\frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)+q}}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_2& ={\int }_{{\left(E_1^{1,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_i(q-1)+}{}^q}\\ & ⪯d_{1,R}^{-\left[\right({\gamma }_1+1)+q]}\cdot mes\left(E_1^{1,\pm }\right)⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ I_{n,1}^{1,\pm }\left(z\right)& ={\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_1+{\left[I_{n,1}^{1,\pm }\right(z\left)\right]}_2⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_2^{1,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_1|}^{{\gamma }_1(q-1)+}{}^q}⪯{\int }_{c_1{d}_{1,R}}^{\left|l_1^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_1(q-1)+q}}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_2^{1,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_1(q-1)+}{}^q}⪯d_{1,R}^{-({\gamma }_1+1)(q-1)};\\ I_{n,2}^{1,\pm }\left(z\right)& ={\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_1+{\left[I_{n,2}^{1,\pm }\right(z\left)\right]}_2⪯d_{1,R}^{-({\gamma }_1+1)(q-1)}.\end{array}$(25) ) and (26(26) $\begin{array}{rl}{\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_1^{2,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)+q}}\\ & ⪯{\int }_0^{c_2{d}_{2,R}}\frac{ds}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_1^{2,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_2(q-1)+q}}\\ & ⪯d_{2,R}^{-\left[\right({\gamma }_2+1)+q]}\cdot mes\left(E_1^{2,\pm }\right)⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ I_{n,1}^{2,\pm }\left(z\right)& ={\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_1+{\left[I_{n,1}^{2,\pm }\right(z\left)\right]}_2⪯d_{2,R}^{-({\gamma }_1+1)(q-1)};\\ {\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_1& ⪯{\int }_{{\left(E_2^{2,\pm }\right)}_1}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z_2|}^{{\gamma }_2(q-1)+q}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ {\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_2& ⪯{\int }_{{\left(E_2^{2,\pm }\right)}_2}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^{{\gamma }_2(q-1)+q}}⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^{{\gamma }_2(q-1)+q}}⪯d_{2,R}^{-({\gamma }_2+1)(q-1)};\\ I_{n,2}^{2,\pm }\left(z\right)& ={\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_1+{\left[I_{n,2}^{2,\pm }\right(z\left)\right]}_2⪯d_{2,R}^{-({\gamma }_1+1)(q-1)}.\end{array}$(26) ), we obtain (44) $\begin{array}{r}\begin{array}{rl}& I_{n,1}^{1,\pm }\left(z\right)⪯{\int }_{E_1^{1,\pm }}{|\zeta -z_1|}^{(-{\gamma }_1)(q-1)}\left|\mathrm{d}\zeta \right|⪯d_{1,R}^{(-{\gamma }_1)(q-1)}mes{E}_1^1⪯1,\\ & I_{n,2}^{1,\pm }\left(z\right)⪯{\int }_{E_2^{1,\pm }}{|\zeta -z_1|}^{(-{\gamma }_1)(q-1)}\left|\mathrm{d}\zeta \right|⪯\left|l_1^{\pm }\right|{}^{(-{\gamma }_1)(q-1)+1}⪯1,\\ & I_{n,1}^1\left(z\right)+I_{n,2}^1\left(z\right)⪯1;\end{array}\end{array}$(44) and (45) $\begin{array}{r}\begin{array}{rl}& I_{n,1}^{2,\pm }\left(z\right)⪯{\int }_{E_1^{2,\pm }}{|\zeta -z_2|}^{(-{\gamma }_2)(q-1)}\left|\mathrm{d}\zeta \right|⪯d_{2,R}^{(-{\gamma }_2)(q-1)}mes{E}_1^{2,\pm }⪯1,\\ & I_{n,2}^{2,\pm }\left(z\right)⪯{\int }_{E_2^{2,\pm }}{|\zeta -z_2|}^{(-{\gamma }_2)(q-1)}\left|\mathrm{d}\zeta \right|⪯\left|l_2^{\pm }\right|{}^{(-{\gamma }_2)(q-1)+1}⪯1,\\ & I_{n,1}^{2,\pm }\left(z\right)+I_{n,2}^{2,\pm }\left(z\right)⪯1.\end{array}\end{array}$(45) Therefore, in this case, from (24(24) $\begin{array}{r}{A}_n\left(z\right)⪯{\Vert P_n\Vert }_p\sum _{i=1}^2{\left[I_{n,1}^{i,\pm }\right(z)+I_{n,2}^{i,\pm }(z\left)\right]}^{\frac{1}{q}},i=1,2.\end{array}$(24) ) to (28(28) $\begin{array}{r}\begin{array}{rl}{I}_{n,1}^{2,\pm }\left(z\right)& ⪯{\int }_{E_1^{2,\pm }}\frac{{|\zeta -z_2|}^{-{\gamma }_2(q-1)}}{|\zeta -z|{}^q}\left|\mathrm{d}\zeta \right|\\ & ⪯d_{2,R}^{(-{\gamma }_2)(q-1)}mes{E}_1^{2,\pm }⪯d_{2,R}^{(-{\gamma }_2-1)(q-1)};\\ I_{n,2}^{2,\pm }\left(z\right)& ⪯{\int }_{E_2^{2,\pm }}\frac{{|\zeta -z_2|}^{-{\gamma }_2(q-1)}}{|\zeta -z|{}^q}\left|\mathrm{d}\zeta \right|⪯{\int }_{c_2{d}_{2,R}}^{\left|l_2^{\pm }\right|}\frac{\mathrm{d}s}{s^q}⪯d_{2,R}^{-(q-1)}.\end{array}\end{array}$(28) ), we obtain (46) $\begin{array}{rl}{A}_n\left(z\right)& ⪯{\Vert P_n\Vert }_p\{\begin{array}{ll}{d}_{1,R}^{1-{\displaystyle \frac{{\gamma }_1+1}{p}}}+d_{2,R}^{1-{\displaystyle \frac{{\gamma }_2+1}{p}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln {\displaystyle \frac{1}{d_{1,R}}})}^{1-{\displaystyle \frac{1}{p}}}+{(\ln {\displaystyle \frac{1}{d_{2,R}}})}^{1-{\displaystyle \frac{1}{p}}},& {\gamma }_1,{\gamma }_2=p-1,\\ 1,& {\gamma }_1,{\gamma }_2<p-1,\end{array}\end{array}$(46) (47) $\begin{array}{rl}{B}_n\left(z\right)& ={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}⪯{\int }_L\left|\mathrm{d}\zeta \right|⪯1.\end{array}$(47) Comparing (19(19) $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& \le \frac{\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|}{2\pi }\\ & \times [{\int }_L\left|\frac{P_n\left(\zeta \right)}{{\mathrm{\Phi }}^{n+1}\left(\zeta \right)}\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{\left|{\mathrm{\Phi }}^{n+1}\right(\zeta \left)\right|{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}]\\ & ⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{1}{d(z,L)}{\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|}+\left|P_n\left(z\right)\right|{\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}],\end{array}$(19) ), (20(20) $\begin{array}{r}{A}_n\left(z\right):={\int }_L\frac{\left|P_n\left(\zeta \right)\right|\left|\mathrm{d}\zeta \right|}{|\zeta -z|};B_n\left(z\right):={\int }_L\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2},\end{array}$(20) ), (30(30) $\begin{array}{r}{B}_n\left(z\right)=\sum _{i,k=1}^2{\int }_{E_k^{i,+}\cup E_k^{i,-}}\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}=:M\left(E_k^{i,+}\right)+M\left(E_k^{i,-}\right).\end{array}$(30) ) and (31(31) $\begin{array}{r}{B}_n\left(z\right)⪯\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}}.\end{array}$(31) ), we have (48) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\left|{\mathrm{\Phi }}^{n+1}\right(z\left)\right|[\frac{{\Vert P_n\Vert }_p}{d(z,L)}{B}_{n,1}+\left|P_n\left(z\right)\right|],\end{array}$(48) where $B_{n,1}$ is defined as in (39(39) $\begin{array}{r}{B}_{n,1}\approx \{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(39) ).

According to (39(39) $\begin{array}{r}{B}_{n,1}\approx \{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(39) ), from (48(49) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p{B}_{n,1}.\end{array}$(49) ), we have (49) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p{B}_{n,1}.\end{array}$(49) Therefore, we complete the proof of the points $z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right)$,  and hence, the proof of Theorem 2.1.

Proof

Proof of Corollaries 2.1 and 2.2

(1) Let $p>1;G\in PDS({\lambda }_1;{\lambda }_2)$for some$0<{\lambda }_1,{\lambda }_2\le 2;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for j = 2. From (35(35) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}})+B_{n,1}(\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}})].\end{array}$(35) ), (34(34) $\begin{array}{r}{B}_{n,1}:=\{\begin{array}{ll}{d}_{1,R}^{-({\displaystyle \frac{{\gamma }_1+1}{p}}-1)}+d_{2,R}^{-(\frac{{\gamma }_2+1}{p}-1)},& {\gamma }_1,{\gamma }_2>p-1,\\ \ln {\displaystyle \frac{1}{d_{1,R}}}+\ln {\displaystyle \frac{1}{d_{2,R}}},& {\gamma }_1={\gamma }_2=p-1,\\ 1,& {\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(34) ), and (36(36) $\begin{array}{r}{d}_{1,R}⪰n^{-{\tilde{\lambda }}_1}.\end{array}$(36) ), for the points $z\in {\mathrm{\Omega }}_R\left(\delta \right),$ we obtain $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(n^{\frac{({\gamma }_1^{\ast }+1)}{p}{\tilde{\lambda }}_1}+n^{\frac{({\gamma }_2^{\ast }+1)}{p}{\tilde{\lambda }}_2})+B_{n,1}(n^{{\tilde{\lambda }}_1}+n^{{\tilde{\lambda }}_2})]\\ & ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\\ & \times [n^{\frac{({\hat{\gamma }}^{\ast }+1)}{p}\hat{\lambda }}+\{\begin{array}{ll}{n}^{{\displaystyle \frac{\hat{\gamma }+1}{p}}\hat{\lambda }},& {\gamma }_1,{\gamma }_2>p-1,\\ n^{\hat{\lambda }}{(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ n^{\hat{\lambda }},& -1<{\gamma }_1,{\gamma }_2<p-1,\end{array}]\end{array}$ where $\hat{\lambda }:=max\{{\tilde{\lambda }}_1;{\tilde{\lambda }}_2\},$ ${\hat{\gamma }}^{\ast }:=\{{\gamma }_1^{\ast };{\gamma }_2^{\ast }\};\hat{\gamma }:=\{{\gamma }_1;{\gamma }_2\},$ and for the points $z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),$ from (49(50) $\begin{array}{r}{P}_n^{\mathrm{\prime }}\left(z\right)=\frac{1}{2\pi i}{\int }_{\mathrm{\partial }B(z,d(z,L_R)}\frac{P_n\left(\zeta \right)\mathrm{d}\zeta }{{(\zeta -z)}^2},z\in L.\end{array}$(50) ) and (39(39) $\begin{array}{r}{B}_{n,1}\approx \{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(39) ), we obtain $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\{\begin{array}{ll}{n}^{({\displaystyle \frac{\hat{\gamma }+1}{p}}-1)\hat{\lambda }},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$ (2) Let $p>1;G\in PDS(c{x}^{1+{\alpha }_1};c{x}^{1+{\alpha }_2}),$for some${\alpha }_1,{\alpha }_2>0;h\left(z\right)$ be defined by (1(1) $\begin{array}{r}h\left(z\right):=h_0\left(z\right)\prod _{j=1}^m{|z-z_j|}^{{\gamma }_j},z\in G_{R_0},R_0>1,\end{array}$(1) ) for j = 2. According to (35(35) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(d_{1,R}^{-\frac{({\gamma }_1^{\ast }+1)}{p}}+d_{2,R}^{-\frac{({\gamma }_2^{\ast }+1)}{p}})+B_{n,1}(\frac{1}{d_{1,R}}+\frac{1}{d_{2,R}})].\end{array}$(35) ), (34(34) $\begin{array}{r}{B}_{n,1}:=\{\begin{array}{ll}{d}_{1,R}^{-({\displaystyle \frac{{\gamma }_1+1}{p}}-1)}+d_{2,R}^{-(\frac{{\gamma }_2+1}{p}-1)},& {\gamma }_1,{\gamma }_2>p-1,\\ \ln {\displaystyle \frac{1}{d_{1,R}}}+\ln {\displaystyle \frac{1}{d_{2,R}}},& {\gamma }_1={\gamma }_2=p-1,\\ 1,& {\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(34) ), and (38(38) $\begin{array}{r}{d}_{2,R}⪰n^{-\frac{1}{1+{\alpha }_2}}.\end{array}$(38) ), we obtain $\begin{array}{rl}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|& ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p[(n^{\frac{({\gamma }_1^{\ast }+1)}{p(1+{\alpha }_1)}}+n^{\frac{({\gamma }_2^{\ast }+1)}{p(1+{\alpha }_2)}})+B_{n,1}(n^{\frac{1}{1+{\alpha }_1}}+n^{\frac{1}{1+{\alpha }_2}})]\\ & ⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\\ & \times [n^{\frac{({\hat{\gamma }}_2^{\ast }+1)}{p(1+{\alpha }_{\ast })}}+\{\begin{array}{ll}{n}^{{\displaystyle \frac{\hat{\gamma }+1}{p(1+{\alpha }_{\ast })}}},& {\gamma }_1,{\gamma }_2>p-1,\\ n^{{\displaystyle \frac{1}{1+{\alpha }_{\ast }}}}{(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ n^{\frac{1}{1+{\alpha }_{\ast }}},& -1<{\gamma }_1,{\gamma }_2<p-1,\end{array}]\end{array}$ where ${\alpha }_{\ast }:=\{{\alpha }_1;{\alpha }_2\}$ and, for the point $z\in {\hat{\mathrm{\Omega }}}_R\left(\delta \right),$ from (49(50) $\begin{array}{r}{P}_n^{\mathrm{\prime }}\left(z\right)=\frac{1}{2\pi i}{\int }_{\mathrm{\partial }B(z,d(z,L_R)}\frac{P_n\left(\zeta \right)\mathrm{d}\zeta }{{(\zeta -z)}^2},z\in L.\end{array}$(50) ) and (39(39) $\begin{array}{r}{B}_{n,1}\approx \{\begin{array}{ll}{n}^{({\displaystyle \frac{{\gamma }_1+1}{p}}-1){\tilde{\lambda }}_1}+n^{({\displaystyle \frac{{\gamma }_2+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_2}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1,\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$(39) ), we obtain $\begin{array}{rl}& \left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯\frac{\left|{\mathrm{\Phi }}^{2(n+1)}\right(z\left)\right|}{d(z,L)}{\Vert P_n\Vert }_p\\ & \times \{\begin{array}{ll}{n}^{({\displaystyle \frac{\hat{\gamma }+1}{p}}-1){\displaystyle \frac{1}{1+{\alpha }_{\ast }}}},& {\gamma }_1,{\gamma }_2>p-1,\\ {(\ln n)}^{1-{\displaystyle \frac{1}{p}}},& \{\begin{array}{l}\mathrm{i}\mathrm{f}{\gamma }_1=p-1,-1<{\gamma }_2\le p-1;\\ \mathrm{o}\mathrm{r}-1<{\gamma }_1\le p-1,{\gamma }_2=p-1,\end{array}\\ 1,& -1<{\gamma }_1,{\gamma }_2<p-1.\end{array}\end{array}$

Proof

Proof of Theorem 2.2

Let $G\in PDS({\lambda }_1;c{x}^{1+{\alpha }_2}),$ for some $0<{\lambda }_1\le 2,$ and ${\alpha }_2>0$. Let $z\in L$ be a arbitrary fixed point and $B(z,d(z,L_R):=\{t:|t-z|<d(z,L_R)\}.$ The Cauchy integral representation for the derivatives of $P_n^{\mathrm{\prime }}\left(z\right)$ is given as (50) $\begin{array}{r}{P}_n^{\mathrm{\prime }}\left(z\right)=\frac{1}{2\pi i}{\int }_{\mathrm{\partial }B(z,d(z,L_R)}\frac{P_n\left(\zeta \right)\mathrm{d}\zeta }{{(\zeta -z)}^2},z\in L.\end{array}$(50) Here, we have (51) $\begin{array}{r}\left|P_n^{\mathrm{\prime }}\left(z\right)\right|=\frac{1}{2\pi }{\int }_{\mathrm{\partial }B(z,d(z,L_R)}\left|P_n\left(\zeta \right)\right|\frac{\left|\mathrm{d}\zeta \right|}{{|\zeta -z|}^2}⪯\underset{z\in {\overline{G}}_R}{max}\left|P_n\left(\zeta \right)\right|\underset{z\in L}{sup}\left\{\frac{1}{d(z,L_R)}\right\}.\end{array}$(51) According to [3, Cor.1.3] and applying (5(5) $\begin{array}{r}{\Vert P_n\Vert }_{C\left({\overline{G}}_R\right)}\le R^n{\Vert P_n\Vert }_{C\left(\overline{G}\right)}.\end{array}$(5) ), we have (52) $\begin{array}{rl}& \underset{z\in {\overline{G}}_R}{max}\left|P_n\left(\zeta \right)\right|⪯{\Vert P_n\Vert }_{C\left(\overline{G}\right)}⪯{\Vert P_n\Vert }_p\\ & \times [n^{\frac{({\gamma }_1^{\ast }+1){\tilde{\lambda }}_1}{p}}+\{\begin{array}{ll}{n}^{{\displaystyle \frac{{\gamma }_2^{\ast }+1}{p(1+{\alpha }_2)}}+{\displaystyle \frac{{\alpha }_2}{p(1+{\alpha }_2)}}},& 1<p<2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ (n\ln n{)}^{1-{\displaystyle \frac{1}{p}}},& p=2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{1-{\displaystyle \frac{1}{p}}},& p>2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}}.\end{array}]\end{array}$(52) On the other hand, from (36(36) $\begin{array}{r}{d}_{1,R}⪰n^{-{\tilde{\lambda }}_1}.\end{array}$(36) ) and (38(38) $\begin{array}{r}{d}_{2,R}⪰n^{-\frac{1}{1+{\alpha }_2}}.\end{array}$(38) ), we have $\begin{array}{r}\underset{z\in L}{sup}\left\{\frac{1}{d(z,L_R)}\right\}\approx sup\{\underset{z\in L^1}{sup}\left\{\frac{1}{d(z,L_R)}\right\};\underset{z\in L^2}{sup}\left\{\frac{1}{d(z,L_R)}\right\}\}⪯n^{{\tilde{\lambda }}_1}.\end{array}$ Therefore, $\begin{array}{rl}& \left|P_n^{\mathrm{\prime }}\left(z\right)\right|⪯{\Vert P_n\Vert }_p\\ & \times [n^{(\frac{({\gamma }_1^{\ast }+1)}{p}+1){\tilde{\lambda }}_1}+\{\begin{array}{ll}{n}^{{\displaystyle \frac{{\gamma }_2^{\ast }+1}{p(1+{\alpha }_2)}}+{\displaystyle \frac{{\alpha }_2}{p(1+{\alpha }_2)}}+{\tilde{\lambda }}_1},& 1<p<2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{{\tilde{\lambda }}_1+1-{\displaystyle \frac{1}{p}}}(\ln n{)}^{1-{\displaystyle \frac{1}{p}}},& p=2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{{\tilde{\lambda }}_1+1-{\displaystyle \frac{1}{p}}},& p>2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\end{array}]\end{array}$ and the proof of Theorem 2.2 is complete.

4.1. Proof of remark

Proof.

The proof of the accuracy of inequalities (14(14) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_4{\Vert P_n\Vert }_p[F_{n,1}+F_{n,2}],\end{array}$(14) )–(16(16) $\begin{array}{r}{\Vert P_n^{\mathrm{\prime }}\Vert }_{\mathrm{\infty }}\le c_6{\Vert P_n\Vert }_p\{\begin{array}{ll}{n}^{{\displaystyle \frac{{\hat{\gamma }}^{\ast }+1}{p(1+{\alpha }_{\ast })}}+{\displaystyle \frac{{\alpha }^{\ast }}{p(1+{\alpha }_{\ast })}}+1},& 1<p<2+{\displaystyle \frac{{\hat{\gamma }}^{\ast }}{1+{\alpha }_2}},\\ n^{2-{\displaystyle \frac{1}{p}}}(\ln n{)}^{1-{\displaystyle \frac{1}{p}}},& p=2+{\displaystyle \frac{{\gamma }_1^{\ast }}{1+{\alpha }_2}}=2+{\displaystyle \frac{{\gamma }_2^{\ast }}{1+{\alpha }_2}},\\ n^{2-{\displaystyle \frac{1}{p}}},& p>2+{\displaystyle \frac{{\hat{\gamma }}^{\ast }}{1+{\alpha }_2}},\end{array}\end{array}$(16) ) can be divided into two parts: (i) $\Vert P_n^{\mathrm{\prime }}{\Vert }_{\mathrm{\infty }}⪯n\Vert P_n{\Vert }_{\mathrm{\infty }};$ (ii) $\Vert P_n{\Vert }_{\mathrm{\infty }}⪯{\mu }_n\Vert P_n{\Vert }_p.$ The first inequality is the well-known sharp Markov inequality. The sharpness of inequality (ii) can be verified in the following examples (see, [[16],[17, Rm 2.9]]). Let $T_n\left(z\right)=1+z+\cdots +z^n$, $h^{\ast }\left(z\right)=$ $h_0\left(z\right),$ $h^{\ast \ast }\left(z\right)=|z-1{|}^{\gamma },\gamma >0,$ $L:=\{z:|z|=1\}.$ Then, for any $n\in \mathbb{N},$ there exist $c_1=c_1(h^{\ast },p)>0$, $c_2=c_2(h^{\ast \ast },p)>0$ such that

1. ${\Vert T\Vert }_{\mathrm{\infty }}\ge c_1{n}^{\frac{1}{p}}{\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast },L)},p>1$;

2. ${\Vert T\Vert }_{\mathrm{\infty }}\ge c_2{n}^{\frac{\gamma +1}{p}}{\Vert T\Vert }_{{\mathcal{L}}_p(h^{\ast \ast },L)},p>\gamma +1$.

5. Conclusion

So, for derivatives of arbitrary algebraic polynomial $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ in regions with corners (including cusp points), the following estimates are found: (a) at the exterior points $z\in \mathrm{\Omega }=C\overline{G}$ (Theorem 2.1 and Corollaries 2.1–2.2); (b) at the closed region $\overline{G}$ (Theorem 2.2 and Corollaries 2.3–2.4). Based on these estimates, find the possible growth of the derivatives of the algebraic polynomial $\left|P_n^{\mathrm{\prime }}\right(z\left)\right|$ in the whole complex plane (Theorem 2.3). Additionally, the case is given when the area has only one type of corner (Corollaries 2.1–2.4).

Acknowledgments

The author thanks the reviewers for their valuable comments and advice that helped improve this work.

Disclosure statement

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