Majorization problem for general family of functions with bounded radius rotations

Abstract

In this article, we study the majorization problem for the general class of functions with bounded radius rotation, which the authors introduced here. Furthermore, the coefficient bound for majorized functions associated with this class is derived. Consequently, we present new results as corollaries and point out relevant connections between the main results obtained from the ones in the literature.

Keywords:

• Function of bounded radius rotation
• majorization
• starlike function
• subordination

MSC Classification:

• 30C45
• 30C80

1. Introduction and preliminaries

Let $\mathcal{A}$ denote the class of all analytic functions f in the open unit disc $\mho :=\left\{t\in \mathbb{C}:\left|t\right|<1\right\}$ satisfying the normalized condition $f(0)=0=f^{\prime }(0)-1$ whose series representation is given as: (1) $$f(t)=t+\sum _{n=2}^{\infty }{a}_n{t}^n,\hspace{1em}t\in \mho .$$(1)

Let $\mathcal{S}\subset \mathcal{A}$ be the class of all univalent functions in $\mho .$ A function f is subordinate to a function $g\in \mathcal{A}$ if there exists an analytic function $\omega :\mho \to \mho$ with $\omega (0)=0,$ such that $f(t)=g(t)°\omega (t),t\in \mho .$ In addition, if g is univalent in $\mho ,$ then $f(t)\prec g(t)$ if and only if $f(0)=g(0)$ and $f(\mho )\subset g(\mho ).$ Let f have the form (1)(1) $$f(t)=t+\sum _{n=2}^{\infty }{a}_n{t}^n,\hspace{1em}t\in \mho .$$(1) and ${\mathcal{S}}_f(t)$ be defined by ${\mathcal{S}}_f(t)=t{f}^{\prime }(t)/f(t).$ The class ${\mathcal{S}}^{\ast }$ of all starlike functions in $\mho$ was one of the earliest subfamilies of $\mathcal{S}.$ Functions $f\in {\mathcal{S}}^{\ast }$ are characterized analytically with the condition $\text{Re}({\mathcal{S}}_f(t))>0$ in $\mho .$ This class was first comprehensively introduced and studied by Shanmugam (1989) using the properties of convolution and subordination, and later, coefficients-related results of this general family were established by Ma and Minda (1992). For this general class, they considered analytic function $\varphi$ with the following properties:

1. $\varphi \in \mathcal{S}$ with $\text{Re}\varphi (t)>0$ in $\mho ;$

2. $\varphi$ is starlike with respect to $\varphi (0)=1;$

3. $\varphi (\mho )$ is symmetric about the real axis, and

4. ${\varphi }^{\prime }(0)>0.$

Hence, they introduced the class (2) $${\mathcal{S}}^{\ast }(\varphi )=\left\{f\in \mathcal{A}:{\mathcal{S}}_f(t)\prec \varphi (t),\hspace{1em}t\in \mho \right\}.$$(2)

Consider the function ${\varphi }_j:\mho \to \mathbb{C}$ defined by ${\varphi }_1(t)={\mathrm{e}}^t,{\varphi }_2(t)={(1+\mathit{\text{st}})}^2,s\in (0,1/\sqrt{2}],{\varphi }_3(t)=t+\sqrt{1+t^2}$ and ${\varphi }_4(t)=(1+\mathit{\text{At}})/(1+\mathit{\text{Bt}}),-1\le B<A\le 1,$ which reduces to ${\varphi }_5(t)=(1+t)/(1-t)$ for $A=1,B=-1$ and ${\varphi }_6(t)=(1+(1-2\alpha )t)/(1-t),\alpha \in [0,1)$ for $A=1-2\alpha ,B=-1.$ For $\varphi ={\varphi }_j (j=1,2,\dots ,6),$ the class ${\mathcal{S}}^{\ast }(\varphi )$ becomes the classes ${\mathcal{S}}_{\mathrm{e}}^{\ast },{\mathcal{S}}_L^{\ast }(s),{\mathcal{S}}_C^{\ast },{\mathcal{S}}^{\ast }(A,B),{\mathcal{S}}^{\ast }$ and ${\mathcal{S}}_{\alpha }^{\ast },$ respectively. The geometric properties of these families were discussed by Janowski (1973), Masih and Kanas (2020), Mendiratta, Nagpal, and Ravichandran (2015), Raina and Sokół (2015), and Robertson (1936).

A natural extension of the class ${\mathcal{S}}^{\ast }$ is the class ${\mathcal{S}}_{\kappa }^{\ast }$ of functions with bounded radius rotaion. Geometrically speaking, $f\in {\mathcal{S}}_{\kappa }^{\ast }$ means that the total variation of angle between radius vector $f(r{e}^{i\theta })$ makes with positive real axis is bounded by $\kappa \pi .$ Also, $f\in {\mathcal{S}}_{\kappa }^{\ast }$ if and only if for $\kappa \ge 2,$ it has a representation: $${\mathcal{S}}_f(t)=\left(\frac{\kappa +2}{4}\right)p_1(t)-\left(\frac{\kappa -2}{4}\right)p_2(t),$$ where $p_j(t)\prec {\varphi }_5(t)(j=1,2)$ (Pinchuk, 1971). Recently, this class received attention in the direction of Ma and Minda’s class. Afis and Noor (2020) and Jabeen and Saliu (2022) introduced the classes ${\mathcal{S}}_{C_{\kappa }}^{\ast }$ and ${\mathcal{S}}_{L_{\kappa }}^{\ast }(s)$ of bounded radius rotation related to crescent and limaçon domains. They investigated inclusion properties, radius problems, coefficient inequalities, and other associated geometrical properties. For more information on the study of ${\mathcal{S}}_{\kappa }^{\ast }$ and its subfamilies, one may see “A survey on functions of bounded boundary and bounded radius rotation” by Noor and Malik§ (2012).

Let $f,g\in \mathcal{A}.$ We say that f is majorized by g or g majorized f (written as $f(t)\ll g(t)$) in $\mho$ if $|f(t)|\le |g(t)|$ for all $t\in \mho .$ The study of this concept started with the work of MacGregor (1967), where he showed that if $f(t)\ll g(t)$ in $\mho ,$ then there exists an analytic function $\phi$ with $|\phi (t)|\le 1$ in $\mho$ such that $$f(t)=\phi (t)g(t),\hspace{1em}t\in \mho .$$

This article opened the door to study the majorization problems for various classes of univalent functions (see Adegani, Alimohammadi, Bulboacă, & Cho, 2022; Cho, Oroujy, Analouei Adegani, & Ebadian, 2020; Gangania & Kumar, 2022; Gupta, Nagpal, & Ravichandran, 2021; Hameed Mohammed & Analouei Adegani, 2023). Cho et al. (2020) examined the majorization property for the class ${\mathcal{S}}^{\ast }(\varphi ),$ which was later extended by Hameed Mohammed and Analouei Adegani (2023) and Adegani et al. (2022).

Motivated by the recent work (Afis & Noor, 2020; Cho et al., 2020; Jabeen & Saliu, 2022), we introduce and investigate the majorization problem for the general class ${\mathcal{S}}_{\kappa }^{\ast }(\varphi )$ of functions of bounded radius rotation. Furthermore, we obtain the coefficient inequality for majorized functions associated with this class and illustrate some particular cases of our results.

Definition 1.1.

Let $f\in \mathcal{A}.$ Then $f\in {\mathcal{S}}_{\kappa }^{\ast }(\varphi )$ if and only for $\kappa \ge 2,$ $${\mathcal{S}}_f(t)=\left(\frac{\kappa +2}{4}\right)p_1(t)-\left(\frac{\kappa -2}{4}\right)p_2(t),$$ where $p_j(t)\prec \varphi (t)(j=1,2).$

As a special case, we have the following:

1. For $\varphi (t)={\varphi }_2(t),$ we have the class introduced and studied by Jabeen and Saliu (2022).

2. For $\varphi (t)={\varphi }_3(t),$ we have the class introduced and studied by Afis and Noor (2020).

3. For $\varphi (t)={\varphi }_4(t),$ we obtain the class introduced and studied by Noor (2009).

4. For $\kappa =2,$ we arrive at the general family of starlike functions introduced and investigated by Ma and Minda (1992).

Remark 1.

When we replace $\varphi (t)$ with Ma and Minda functions, we obtain the corresponding subclasses of bounded radius rotation of Ma and Minda type.

Lemma 1.2.

[Kanas & Sugawa, 2006, Proposition 5.4] If $\varphi$ is convex univalent, has non-negative Taylor coefficients about the origin and $\text{Re}(\varphi (t))>0$ in $\mho ,$ then $$\underset{\left|t\right|=r}{\text{min}}\left|\varphi \right(t\left)\right|=\varphi (-r)\le \left|\varphi \right(t\left)\right|\le \varphi \left(r\right)=\underset{\left|t\right|=r}{\text{max}}\left|\varphi \right(t\left)\right|.$$

2. Main results

Theorem 2.1.

Let $f\in \mathcal{A}$ be majorized by $g\in {\mathcal{S}}_{\kappa }^{\ast }(\varphi ).$ Then $g^{\prime }(t)$ majorized $f^{\prime }(t)$ in the disc $\left|t\right|<r_{\kappa },$ where $r_{\kappa }$ is the smallest positive root of the equation: (3) $$(1-r^2)\left|(\kappa +2)\underset{\left|t\right|=r}{\text{min}}\left|\varphi \right(t\left)\right|-(\kappa -2)\underset{\left|t\right|=r}{\text{max}}\left|\varphi \right(t\left)\right|\right|-8r=0.$$(3)

Proof.

Since $f(t)\ll g(t),$ then $f(t)=\phi (t)g(t)$ for some analytic function $\phi$ with $|\phi (t)|\le 1,t\in \mho .$ Therefore, (4) $$\begin{array}{c}{f}^{\prime }(t)={\phi }^{\prime }(t)g(t)+\phi (t)g^{\prime }(t)\\ =\left({\phi }^{\prime }(t)\frac{g(t)}{g^{\prime }(t)}+\phi (t)\right)g^{\prime }(t).\end{array}$$(4)

But $g\in {\mathcal{S}}_{\kappa }^{\ast }(\varphi )$ implies that (5) $$\begin{array}{l}{\mathcal{S}}_g(t)=\left(\frac{\kappa +2}{4}\right)p_1(t)-\left(\frac{\kappa -2}{4}\right)p_2(t),\hspace{1em}{p}_j(t)\prec \varphi (t)(j=1,2)\\ =\left(\frac{\kappa +2}{4}\right)\varphi ({\omega }_1(t))-\left(\frac{\kappa -2}{4}\right)\varphi ({\omega }_2(t)),\end{array}$$(5) where ${\omega }_j$ (j = 1, 2) is analytic in $\mho$ with ${\omega }_j(0)=0$ such that $|{\omega }_j(t)|=r<1.$ Since $\text{Re}(\varphi ({\omega }_j(t)))>0 (j=1,2)$ in $\mho ,$ we can rewrite (5)(5) $$\begin{array}{l}{\mathcal{S}}_g(t)=\left(\frac{\kappa +2}{4}\right)p_1(t)-\left(\frac{\kappa -2}{4}\right)p_2(t),\hspace{1em}{p}_j(t)\prec \varphi (t)(j=1,2)\\ =\left(\frac{\kappa +2}{4}\right)\varphi ({\omega }_1(t))-\left(\frac{\kappa -2}{4}\right)\varphi ({\omega }_2(t)),\end{array}$$(5) as (6) $$\frac{g(t)}{g^{\prime }(t)}=\frac{t}{\left(\frac{\kappa +2}{4}\right)\varphi ({\omega }_1(t))-\left(\frac{\kappa -2}{4}\right)\varphi ({\omega }_2(t))}.$$(6)

From the minimum and maximum modulus principle, we have that $$\underset{\left|t\right|=r}{\text{min}}\left|\varphi \right(t\left)\right|\le \underset{\left|{\omega }_1\right(t\left)\right|=r}{\text{min}}\left|\varphi \right({\omega }_1\left(t\right)\left)\right|=\underset{\left|{\omega }_1\right(t\left)\right|\le r}{\text{min}}\left|\varphi \right({\omega }_1\left(t\right)\left)\right|$$ and $$\underset{\left|{\omega }_2\right(t\left)\right|\le r}{\text{max}}\left|\varphi \right({\omega }_2\left(t\right)\left)\right|=\underset{\left|{\omega }_2\right(t\left)\right|=r}{\text{max}}\left|\varphi \right({\omega }_2\left(t\right)\left)\right|\le \underset{\left|t\right|=r}{\text{max}}\left|\varphi \right(t\left)\right|.$$

In view of these inequalities, (6)(6) $$\frac{g(t)}{g^{\prime }(t)}=\frac{t}{\left(\frac{\kappa +2}{4}\right)\varphi ({\omega }_1(t))-\left(\frac{\kappa -2}{4}\right)\varphi ({\omega }_2(t))}.$$(6) becomes (7) $$\begin{array}{l}\left|\frac{g\left(t\right)}{g^{\prime }\left(t\right)}\right|\le \frac{r}{|\left(\frac{\kappa +2}{4}\right)\left|\varphi \left({\omega }_1\right(t\left)\right)\right|-\left(\frac{\kappa -2}{4}\right)\left|\varphi \left({\omega }_2\right(t\left)\right)\right||}\\ \le \frac{4r}{|(\kappa +2)\underset{\left|{\omega }_1\right(t\left)\right|=r}{\text{min}}\left|\varphi \left({\omega }_1\right(t\left)\right)\right|-(\kappa -2)\underset{\left|{\omega }_2\right(t\left)\right|=r}{\text{max}}\left|\varphi \left({\omega }_2\right(t\left)\right)\right||}\\ \le \frac{4r}{|(\kappa +2)\underset{\left|t\right|=r}{\text{min}}\left|\varphi \left(t\right)\right|-(\kappa -2)\underset{\left|t\right|=r}{\text{max}}\left|\varphi \left(t\right)\right||}.\end{array}$$(7)

From (4)(4) $$\begin{array}{c}{f}^{\prime }(t)={\phi }^{\prime }(t)g(t)+\phi (t)g^{\prime }(t)\\ =\left({\phi }^{\prime }(t)\frac{g(t)}{g^{\prime }(t)}+\phi (t)\right)g^{\prime }(t).\end{array}$$(4) and (7)(7) $$\begin{array}{l}\left|\frac{g\left(t\right)}{g^{\prime }\left(t\right)}\right|\le \frac{r}{|\left(\frac{\kappa +2}{4}\right)\left|\varphi \left({\omega }_1\right(t\left)\right)\right|-\left(\frac{\kappa -2}{4}\right)\left|\varphi \left({\omega }_2\right(t\left)\right)\right||}\\ \le \frac{4r}{|(\kappa +2)\underset{\left|{\omega }_1\right(t\left)\right|=r}{\text{min}}\left|\varphi \left({\omega }_1\right(t\left)\right)\right|-(\kappa -2)\underset{\left|{\omega }_2\right(t\left)\right|=r}{\text{max}}\left|\varphi \left({\omega }_2\right(t\left)\right)\right||}\\ \le \frac{4r}{|(\kappa +2)\underset{\left|t\right|=r}{\text{min}}\left|\varphi \left(t\right)\right|-(\kappa -2)\underset{\left|t\right|=r}{\text{max}}\left|\varphi \left(t\right)\right||}.\end{array}$$(7) , we obtain $$|f^{\prime }(t)|\le \left(\frac{4r|{\phi }^{\prime }(t)|}{|\left(\kappa +2\right)\underset{\left|t\right|=r}{\mathrm{\text{min}}}\left|\varphi (t)\right|-\left(\kappa -2\right)\underset{\left|t\right|=r}{\mathrm{\text{max}}}\left|\varphi (t)\right||}+|\phi (t)|\right)|g^{\prime }(t)|.$$

Let $|\phi (t)|=\rho <1,$ and using the Schwarz Pick lemma [Nehari, 1952, p. 168], we arrive at $$|f^{\prime }(t)|\le \left(\frac{4r(1-{\rho }^2)}{(1-r^2)|\left(\kappa +2\right)\underset{\left|t\right|=r}{\mathrm{\text{min}}}\left|\varphi (t)\right|-\left(\kappa -2\right)\underset{\left|t\right|=r}{\mathrm{\text{max}}}\left|\varphi (t)\right||}+\rho \right)|g^{\prime }(t)|.$$

Let $$\mathcal{V}(r,\rho )=\frac{4r(1-{\rho }^2)}{(1-r^2)|\left(\kappa +2\right)\underset{\left|t\right|=r}{\mathrm{\text{min}}}\left|\varphi (t)\right|-\left(\kappa -2\right)\underset{\left|t\right|=r}{\mathrm{\text{max}}}\left|\varphi (t)\right||}+\rho ,\hspace{1em}r\in (0,1),\rho \in [0,1].$$

To find $r_{\kappa },$ we choose $r_{\kappa }=\mathrm{\text{max}}\left\{r\in (0,1):\mathcal{V}(r,\rho )\le 1,\rho \in [0,1]\right\}.$ But $\mathcal{V}(r,\rho )\le 1$ if and only if $$0\le (1-r^2)|\left(\kappa +2\right)\underset{\left|t\right|=r}{\mathrm{\text{min}}}\left|\varphi (t)\right|-\left(\kappa -2\right)\underset{\left|t\right|=r}{\mathrm{\text{max}}}\left|\varphi (t)\right||-4r(1+\rho ):={\mathcal{W}}_{\kappa }(r,\rho ).$$

It is easy to see that ${\mathcal{W}}_{\kappa }(r,\rho )$ is decreasing on $0\le \rho \le 1.$ Thus,

$\mathrm{\text{min}}{\mathcal{W}}_{\kappa }(r,\rho )={\mathcal{W}}_{\kappa }(r,1):={\mathcal{W}}_{\kappa }(r),$ where $${\mathcal{W}}_{\kappa }(r)=(1-r^2)|\left(\kappa +2\right)\underset{\left|t\right|=r}{\mathrm{\text{min}}}\left|\varphi (t)\right|-\left(\kappa -2\right)\underset{\left|t\right|=r}{\mathrm{\text{max}}}\left|\varphi (t)\right||-8r.$$

We note that ${\mathcal{W}}_{\kappa }(0)>0\text{and}{\mathcal{W}}_{\kappa }(1)<0.$ Therefore, there exists $r=r_{\mathit{\kappa }}$ which satisfies (3)(3) $$(1-r^2)\left|(\kappa +2)\underset{\left|t\right|=r}{\text{min}}\left|\varphi \right(t\left)\right|-(\kappa -2)\underset{\left|t\right|=r}{\text{max}}\left|\varphi \right(t\left)\right|\right|-8r=0.$$(3) . Hence, ${\mathcal{W}}_{\kappa }(r)\ge 0$ for all $r$ in the disc $\left|t\right|<r_{\kappa }.$ □

It is worthy of note that for $\kappa =2,$ Theorem 2.1 reduces to Theorem 2 in Cho et al. (2020) and also becomes various corollaries obtained therein for respective Ma and Minda functions. Now for $\kappa >2,$ we obtain the following results.

Corollary 2.2.

Let $f\in \mathcal{A}\text{and}f(t)\ll g(t)\text{with}g\in {\mathcal{S}}_{\kappa }^{\ast }({\varphi }_6):={\mathcal{S}}_{\kappa }^{\ast }(\alpha ).$ Then $f^{\prime }(t)\ll g^{\prime }(t)$ in the disc $\left|t\right|<r_{\kappa }(\alpha ),$ where $$r_{\kappa }(\alpha )=\{\begin{array}{cc}\hfill & \frac{2}{k(1-\alpha )+2+\sqrt{{(1-\alpha )}^2{\kappa }^2+4(1-\alpha )\kappa +8\alpha }}\hspace{1em}\alpha \ne \frac{1}{2}\hfill \\ \hfill & \frac{2}{\kappa +2},\hspace{1em}\alpha =\frac{1}{2}\hfill \end{array}$$ or $$r_{\kappa }(\alpha )=\{\begin{array}{cc}\hfill & \frac{2}{k(1-\alpha )-2+\sqrt{{(1-\alpha )}^2{\kappa }^2-4(1-\alpha )\kappa +8\alpha }}\hspace{0.17em}\alpha \ne \frac{1}{2}\hspace{0.17em}\text{and}\hspace{0.17em}0\le \alpha \le \frac{\kappa -4}{\kappa -2}\hspace{0.17em}\text{with}\hspace{0.17em}\kappa >4\hfill \\ \hfill & \frac{2}{\kappa +2},\hspace{1em}\alpha =\frac{1}{2}\hfill \end{array}$$

Proof.

Since ${\varphi }_6(t)$ is convex in $\mho ,$ then by Lemma 1.2 and Theorem 2.1, we have the required result. □

For $\alpha =0$ in Corollary 2.2, we arrive at the following result.

Corollary 2.3.

Let $f\in \mathcal{A}\text{and}f(t)\ll g(t)\text{with}g\in {\mathcal{S}}_{\mathit{\kappa }}^{\ast }({\varphi }_5):={\mathcal{S}}_{\kappa }^{\ast }.$ Then $f^{\prime }(t)\ll g^{\prime }(t)$ in the disc $\left|t\right|<r_{\kappa },$ where $$r_{\kappa }=\frac{2}{k+2+\sqrt{\kappa (\kappa +4)}}$$ or $$r_{\kappa }=\begin{array}{c}\frac{2}{k-2+\sqrt{\kappa (\kappa -4)}}\hspace{0.17em}\hspace{0.17em}\text{with}\hspace{1em}\kappa >4.\hfill \end{array}$$

Corollary 2.4.

Let $f\in \mathcal{A}\text{and}f(t)\ll g(t)\text{with}g\in {\mathcal{S}}_{\kappa }^{\ast }({\varphi }_2):={\mathcal{S}}_{L_{\kappa }}^{\ast }(s).$ Then $g^{\prime }(t)\text{majorized}{f}^{\prime }(t)$ in the disc $\left|t\right|<r_{\kappa }(s),$ where $r_{\kappa }\left(s\right)$ is the smallest roots of the equation $$1-(\kappa s+2)r-(1-s^2)r^2+\mathit{\text{ks}}{r}^3-s^2{r}^4=0$$ or $$-1+(\kappa s+2)r+(1-s^2)r^2-\mathit{\text{ks}}{r}^3+s^2{r}^4=0.$$

Proof.

The method of proof is the same as the proof of Corollary 2.2. □

Corollary 2.5.

Let $f\in \mathcal{A}\text{and}f(t)\ll g(t)\text{with}g\in {\mathcal{S}}_{\kappa }^{\ast }({\varphi }_1):={\mathcal{S}}_{{\mathrm{e}}_{\kappa }}^{\ast }.$ Then $g^{\prime }(t)\text{majorized}{f}^{\prime }(t)$ in the disc $\left|t\right|<r_{\kappa }(s),$ where $r_{\kappa }$ is the smallest roots of the equation $$(1-r^2)\left|4\mathrm{\text{cosh}}(r)-\kappa \mathrm{\text{sinh}}(r)\right|-8r=0.$$

Theorem 2.6.

Let $\varphi (t)=1+B_{1}t+B_2{t}^2+B_3{t}^3+\cdots$ be convex in $\mho$ and $f\in \mathcal{A}$ have the form (1)(1) $$f(t)=t+\sum _{n=2}^{\infty }{a}_n{t}^n,\hspace{1em}t\in \mho .$$(1) . If $f(t)\ll g(t)\text{with}g\in {\mathcal{S}}_{\kappa }^{\ast }(\varphi ),$ then (8) $$\left|a_n\right|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left(\frac{2(j-2)+\kappa \left|B_1\right|}{2(n-1)!}\right)\right].$$(8)

Proof.

Let (9) $${\mathcal{S}}_g(t)=p(t)=1+\sum _{n=1}^{\infty }{d}_n{t}^n.$$(9)

Since $g\in {\mathcal{S}}_{\kappa }^{\ast }(\varphi ),$ then there exist $p_j(t)=1+{\sum }_{n=1}^{\infty }{c}_{n,j}{t}^n\prec \varphi (t),j=1,2$ such that (10) $$1+\sum _{n=1}^{\infty }{d}_n{t}^n=\left(\frac{\kappa +2}{4}\right)\left(1+\sum _{n=1}^{\infty }{c}_{n,1}{t}^n\right)-\left(\frac{\kappa -2}{4}\right)\left(1+\sum _{n=1}^{\infty }{c}_{n,2}{t}^n\right).$$(10)

Comparing the coefficients of $t^n$ in (10)(10) $$1+\sum _{n=1}^{\infty }{d}_n{t}^n=\left(\frac{\kappa +2}{4}\right)\left(1+\sum _{n=1}^{\infty }{c}_{n,1}{t}^n\right)-\left(\frac{\kappa -2}{4}\right)\left(1+\sum _{n=1}^{\infty }{c}_{n,2}{t}^n\right).$$(10) , we arrive at (11) $$\left|d_n\right|\le \left(\frac{\kappa +2}{4}\right)\left|c_{n,1}\right|+\left(\frac{\kappa -2}{4}\right)\left|c_{n,2}\right|.$$(11)

Since $p_j(t)\prec \varphi (t),j=1,2\text{and}\varphi (t)$ is convex, then by Rogosinski’s lemma (Rogosinski, 1945), we have that $\left|c_{n,j}\right|\le \left|B_1\right|,j=1,2.$ Consequently (11)(11) $$\left|d_n\right|\le \left(\frac{\kappa +2}{4}\right)\left|c_{n,1}\right|+\left(\frac{\kappa -2}{4}\right)\left|c_{n,2}\right|.$$(11) becomes $$\left|d_n\right|\le \frac{\kappa }{2}\left|B_1\right|.$$

Let $g(t)=z+{\sum }_{n=2}^{\infty }{b}_n{t}^n.$ Then (9)(9) $${\mathcal{S}}_g(t)=p(t)=1+\sum _{n=1}^{\infty }{d}_n{t}^n.$$(9) implies $$z+\sum _{n=2}^{\infty }n{b}_n{t}^n=\left(z+\sum _{n=2}^{\infty }{b}_n{t}^n\right)\left(1+\sum _{n=1}^{\infty }{d}_n{t}^n\right).$$

Now, following the procedures of Kuroki and Owa (2012), we find that (12) $$\left|b_n\right|\le \prod _{j=2}^n\left(\frac{2(j-2)+\kappa \left|B_1\right|}{2(n-1)!}\right).$$(12)

Let $\phi (t)={\sum }_{n=0}^{\infty }{e}_n{t}^n$ be analytic in $\mho \text{with}|\phi (t)|\le 1\text{in}\mho .$ Then the condition $f(t)\ll g(t)\text{in}\mho$ implies that $f(t)=\phi (t)g(t),$ which follows that (13) $$a_n=e_0{b}_n+e_1{b}_{n-1}+e_2{b}_{n-2}+\cdots +e_{n-2}{b}_2+e_{n-1}.$$(13)

Let $\Gamma$ be any circle in $\left|t\right|=r,(0<r<1).$ Then by Cauchy derivative formula, we can write (13)(13) $$a_n=e_0{b}_n+e_1{b}_{n-1}+e_2{b}_{n-2}+\cdots +e_{n-2}{b}_2+e_{n-1}.$$(13) as $$\begin{array}{l}\left|a_n\right|=\left|\frac{1}{2\pi i}{\int }_{\Gamma }\frac{\phi (t)}{t^n}\left(1+b_{2}t+b_3{t}^2+b_4{t}^3+\cdots +b_n{t}^{n-1}\right) \mathit{\text{dt}}\right|\\ \le \frac{1}{2\pi }{\int }_0^{2\pi }\frac{|\phi (r{\mathrm{e}}^{i\theta })|}{r^{n-1}}\left|1+b_{2}t+b_3{t}^2+b_4{t}^3+\cdots +b_n{t}^{n-1}\right| d\theta \\ \le \frac{1}{r^{n-1}}\left(1+\sum _{m=2}^n\left|b_m\right|\right).\end{array}$$

Since this inequality holds for all $r\in (0,1),$ then by (12)(12) $$\left|b_n\right|\le \prod _{j=2}^n\left(\frac{2(j-2)+\kappa \left|B_1\right|}{2(n-1)!}\right).$$(12) , it follows that $$\begin{array}{l}\left|a_n\right|\le 1+\sum _{m=2}^n\left|b_m\right|\\ \le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left(\frac{2(j-2)+\kappa \left|B_1\right|}{2(n-1)!}\right)\right],\end{array}$$ which conclude the required result. □

Remark 2.

For $\kappa =2,$ Theorem 2.6 becomes Theorem 3 of Cho et al. (2020).

For $\kappa >2,$ we have the following results:

Corollary 2.7.

Let $f\in \mathcal{A}$ and suppose $f(t)\ll g(t)\text{in}\mho \text{with}g\in {\mathcal{S}}_{\kappa }^{\ast }({\varphi }_4):={\mathcal{S}}_{\kappa }^{\ast }(A,B).$ Then $$\left|a_n\right|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left(\frac{2(j-2)+\kappa (A-B)}{2(n-1)!}\right)\right].$$

Proof.

It is easy to see that $${\varphi }_4(t)=1+(A-B)\sum _{n=1}^{\infty }{(-1)}^{n+1}{B}^{n-1}{t}^n.$$

Since ${\varphi }_4$ is a function that maps $\mho$ onto a circular domain and so, it is a convex function. Therefore, by Rogosinski’s lemma (Rogosinski, 1945), $B_1=A-B.$ Thus, in view of Theorem 2.6, we have the required result. □

Corollary 2.8.

Let $f\in \mathcal{A}$ and suppose $f(t)\ll g(t)\text{in}{\mho }_{\frac{\sqrt{2}}{2}}:=\left\{z\in \mathbb{C}:\left|t\right|<\sqrt{2}/2\right\}\text{with}g\in {\mathcal{S}}_{\kappa }^{\ast }({\varphi }_3):={\mathcal{S}}_{C_{\kappa }}^{\ast }.$ Then $$\left|a_n\right|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left(\frac{2(j-2)+\kappa }{2(n-1)!}\right)\right].$$

Proof.

Raina and Sokół (2019) showed that the function ${\varphi }_3$ is convex in the disc ${\mho }_{\frac{\sqrt{2}}{2}}.$ Thus, by Rogosinski’s lemma (Rogosinski, 1945), $B_1=1.$ Consequently, from Theorem 2.6, we obtain the desired result. □

Corollary 2.9.

Let $f\in \mathcal{A}$ and suppose $f(t)\ll g(t)\text{in}\mho \text{with}g\in {\mathcal{S}}_{L_{@@\kappa }}^{\ast }.$ Then for $0<s\le 1/2,$ $$\left|a_n\right|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left(\frac{(j-2)+\kappa s}{(n-1)!}\right)\right].$$

Proof.

We can see that $$\text{Re}\left(\frac{{\left(z{\varphi }_2^{\prime }(t)\right)}^{\prime }}{{\varphi }_2^{\prime }(t)}\right)=\frac{1+2s}{1+s}\ge \frac{1-2s}{1-s}>0\hspace{1em}\text{whenever}\hspace{1em}s\le \frac{1}{2}.$$

This shows that ${\varphi }_2(t)$ is convex in $\mho .$ Therefore, by Rogosinski’s lemma (Rogosinski, 1945), $B_1=s.$ Hence, using Theorem 2.6, we arrive at the desired result. □

Author contributions

All authors contributed equally to this work.

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The authors are thankful to the Heads of their institutions for providing an excellent research environment.

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