Fekete-Szeg\"o Problem For Generalized Bi-Subordinate Functions of Complex Order

TL;DR

This paper establishes a Fekete-Szeg"o inequality for a broad class of generalized bi-subordinate functions of complex order, extending previous results in the field.

Contribution

It introduces a generalized inequality for bi-subordinate functions of complex order, broadening the scope of existing Fekete-Szeg"o results.

Findings

Derived a new Fekete-Szeg"o inequality for generalized bi-subordinate functions.

Generalized previous inequalities to functions of complex order.

Results encompass and extend earlier related works.

Abstract

In this paper, we obtain Fekete-Szeg\"o inequality for the generalized bi-subordinate functions of complex order. The results, which are presented in this study, would generalize those in related works of several earlier authors.

Equations

$$f\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k.$$

$${\mathcal{B}}_0:=\left\{w\in \mathcal{A}:w\left(0\right)=0,\left|w\left(z\right)\right|<1,\left(z\in \mathbb{D}\right)\right\},$$

$$f(z)=g(w(z)),\left(z\in \mathbb{D}\right).$$

$${\mathcal{S}}^{\ast }\equiv \left\{f\in \mathcal{A}:\frac{z{f}^{\prime }(z)}{f(z)}\prec \frac{1+z}{1-z}\right\}$$

$$\mathcal{C}\equiv \left\{f\in \mathcal{A}:1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\prec \frac{1+z}{1-z}\right\}.$$

$${\mathcal{S}}^{\ast }(\phi )\equiv \left\{f\in \mathcal{A}:\frac{z{f}^{\prime }(z)}{f(z)}\prec \phi (z)\right\}$$

$$\mathcal{C}(\phi )\equiv \left\{f\in \mathcal{A}:1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\prec \phi (z)\right\}$$

$${\mathcal{S}}^{\ast }(\gamma ;\phi )\equiv \left\{f\in A:1+\frac{1}{\gamma }\left(\frac{z{f}^{\prime }(z)}{f(z)}-1\right)\prec \phi (z);\gamma \in \mathbb{C}\backslash \left\{0\right\}\right\}$$

$$C(\gamma ;\phi )\equiv \left\{f\in A:1+\frac{1}{\gamma }\left(\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\right)\prec \phi (z);\gamma \in \mathbb{C}\backslash \left\{0\right\}\right\}.$$

$$Q_{1}=\left\{{\begin{array}[]{l}\frac{2\mathcal{B}^{2}}{1-k^{2}};\;\;\;\;\;\;\;\;\;\;\;\;\ \ \ \ 0\leqslant k<1,\\ \frac{8}{\pi^{2}};\;\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\ \ \ \ \ \ \ k=1,\\ \frac{\pi^{2}}{4(k^{2}-1)\sqrt{t}(1+t)\mathcal{K}^{2}(t)};\text{ \ \ \ }k>1,\end{array}}\right.,\text{ \ }Q_{2}=\left\{{\begin{array}[]{l}\frac{(\mathcal{B}^{2}+2)}{3}Q_{1};\;\;\;\;\;\;\;\;\;\;\;\;\ \ \ \ 0\leqslant k<1,\\ \frac{2}{3}Q_{1};\;\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\ \ \ \ \ \ \ \ \ k=1,\\ \frac{\left[{4\mathcal{K}^{2}(t)(t^{2}+6t+1)-\pi^{2}}\right]}{24\sqrt{t}(1+t)\mathcal{K}^{2}(t)}Q_{1};\text{ \ \ \ }k>1,\end{array}}\right.$$

$$\mathcal{S}(\lambda ,\gamma ;\phi )\equiv \left\{f\in A:1+\frac{1}{\gamma }\left(\frac{z{f}^{\prime }(z)+\lambda z^2{f}^{\prime \prime }(z)}{(1-\lambda )f(z)-\lambda z{f}^{\prime }(z)}-1\right)\prec \phi (z)\text{; }0\le \lambda \le 1\text{, }\gamma \in \mathbb{C}\backslash \left\{0\right\}.\right\}$$

$$w(z)=c_{1}z+c_2{z}^2+\cdot \cdot \cdot \left(z\in \mathbb{D}\right),$$

$$\left|c_2-s{c}_1^2\right|\le 1+\left(\left|s\right|-1\right){\left|c_1\right|}^2.$$

$$\left|a_2\right|\le \frac{\left|\gamma \right|B_1}{\left(1+\lambda \right)},$$

$$\left|a_3\right|\le \frac{\left|\gamma \right|\left|B_1\right|}{4\left(1+2\lambda \right)}\max \{2,\left(\left|s\right|+\left|t\right|\right)\}$$

$$\left|a_{3}-\mu a_{2}^{2}\right|\leq\left\{\begin{array}[]{c}\frac{B_{1}\left|\gamma\right|}{2\left(1+2\lambda\right)}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ }\mathcal{L}\leq 2,\\ \frac{B_{1}\left|\gamma\right|}{4\left(1+2\lambda\right)}{L}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ if \ }\mathcal{L}\geq 2,\end{array}\right.$$

$$1+\frac{1}{\gamma }\left(\frac{z{f}^{\prime }(z)+\lambda z^2{f}^{\prime \prime }(z)}{(1-\lambda )f(z)-\lambda z{f}^{\prime }(z)}-1\right)=\phi (u(z))$$

$$=1+\frac{\left(1+\lambda \right)a_2}{\gamma }z+\left[\frac{2\left(1+2\lambda \right)a_3-{\left(1-\lambda \right)}^2{a}_2^2}{\gamma }\right]z^2+\cdot \cdot \cdot$$

$$1+\frac{1}{\gamma }\left(\frac{w{g}^{\prime }(w)+\lambda w^2{g}^{\prime \prime }(w)}{(1-\lambda )g(w)-\lambda w{g}^{\prime }(w)}-1\right)=\phi (v(w))$$

$$=1-\frac{\left(1+\lambda \right)a_2}{\gamma }w+\left[\frac{-2\left(1+2\lambda \right)a_3+\left(3+6\lambda -{\lambda }^2\right)a_2^2}{\gamma }\right]w^2+\cdot \cdot \cdot$$

$$\frac{\left(1+\lambda \right)a_2}{\gamma }$$

$$\frac{-\left(1+\lambda \right)a_2}{\gamma }$$

$$c_1$$

$$a_2$$

$$a_3=a_2^2+\frac{\gamma }{4\left(1+2\lambda \right)}\left[B_1{c}_2+B_2{c}_1^2-B_1{d}_2-B_2{d}_1^2\right],\left|c_k\right|\le 1.$$

$$\left|a_2\right|=\left|\frac{\gamma B_1}{\left(1+\lambda \right)}{c}_1\right|\le \frac{\left|\gamma \right|B_1}{\left(1+\lambda \right)}$$

$$\left|a_3\right|$$

$$\left|a_3\right|\le \frac{\left|\gamma \right|\left|B_1\right|}{4\left(1+2\lambda \right)}\max \{2,\left(\left|s\right|+\left|t\right|\right)\},$$

$$\left|a_3-\mu a_2^2\right|$$

$$\left|a_{3}-\mu a_{2}^{2}\right|\leq\left\{\begin{array}[]{c}\frac{B_{1}\left|\gamma\right|}{2\left(1+2\lambda\right)}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ }\mathcal{L}<2,\\ \frac{B_{1}\left|\gamma\right|}{4\left(1+2\lambda\right)}\mathcal{L}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ if \ }\mathcal{L}\geq 2,\end{array}\right..\text{ }$$

$$\left|a_{3}-\mu a_{2}^{2}\right|\leq\left\{\begin{array}[]{c}\frac{\gamma\left|B_{2}\right|}{2\left(1+2\lambda\right)}-\left(\mu-1\right)\text{ }\frac{\gamma^{2}B_{1}^{2}}{\left(1+\lambda\right)^{2}}\text{\ \ \ \ \ \ if \ }\mu\leq 1,\\ \frac{\gamma\left|B_{2}\right|}{2\left(1+2\lambda\right)}+\left(\mu-1\right)\text{ }\frac{\gamma^{2}B_{1}^{2}}{\left(1+\lambda\right)^{2}}\text{\ \ \ \ \ \ if \ }\mu>1,\end{array}\right.$$

Full text

2010  Mathematics Subject Classification: Primary 30C45; Secondary 30C80.

1. INTRODUCTION

Let $\mathcal{A}$ be the class of analytic functions in the open unit disk $\mathbb{D}=\{z\in\mathbb{\mathbb{C}}:$ $\left|z\right|<1\}$ and let $\mathcal{S}$ be the class of functions $f$ that are analytic and univalent in $\mathbb{D}$ and are of the form

[TABLE]

A function $f\in\mathcal{A}$ is said to be subordinate to a function $g\in\mathcal{A},$ denoted by $f\prec g,$ if there exists a function $w$ $\in\mathcal{B}_{0}$ where

[TABLE]

such that

[TABLE]

We let $\mathcal{S}^{\ast}$ consist of starlike functions $f\in\mathcal{A}$, that is $\Re\{zf^{\prime}(z)\diagup f(z)\}>0$ in $\mathbb{D}$ and $\mathcal{C}$ consist of convex functions $f\in\mathcal{A}$, that is $1+\Re\{zf^{\prime\prime}(z)\diagup f^{\prime}(z)\}>0$ in $\mathbb{D}$. In terms of subordination, these conditions are, respectively, equivalent to

[TABLE]

and

[TABLE]

A generalization of the above two classes, according to Ma and Minda [23], are

[TABLE]

and

[TABLE]

where $\varphi$  is a positive real part function normalized by $\varphi(0)=1$, $\varphi^{\prime}(0)>0$ and $\varphi$ maps $D$ onto a region starlike with respect to $1$ and symmetric with respect to the real axis. Obvious extensions of the above two classes (see [22]) are

[TABLE]

and

[TABLE]

In literature, the functions belonging to these classes are called Ma-Minda starlike and convex of complex order $\gamma$ $\left(\gamma\in\mathbb{C}\diagdown\{0\}\right)$, respectively.

Some of the special cases of the above two classes  $S^{\ast}(\gamma;\varphi)$ and $C(\gamma;\varphi)$ are

• (1)

$\mathcal{S}^{\ast}\left((1,(1+Az)\diagup(1+Bz)\right)=\mathcal{S}[A,B]$ and $\mathcal{C}(1,(1+Az)\diagup(1+Bz))=\mathcal{C}[A,B],$ $\left(-1\leq B<A\leq 1\right)$ are classes of Janowski starlike and convex functions, respectively,

• (2)

$\mathcal{S}^{\ast}((1-\beta)e^{-i\delta}\cos\delta,(1+z)\diagup(1-z))=\mathcal{S}^{\ast}[\delta,\beta]$ and $\mathcal{C}((1-\beta)e^{-i\delta}\cos\delta,(1+z)\diagup(1-z))=\mathcal{C}[\delta,\beta],$ $\left(\left|\delta\right|<\pi\diagup 2,\text{ }0\leq\beta<1\right)$ are classes of $\delta$-spirallike and $\delta$-Robertson univalent functions of order $\beta,$ respectively,

• (3)

$\mathcal{S}^{\ast}(1,(1+(1-2\beta)z)\diagup(1-z))=\mathcal{S}^{\ast}(\beta)$ and $\mathcal{C}\left(1,(1+(1-2\beta)z)\right)\diagup(1-z))=\mathcal{C}(\beta)$ $\left(0\leq\beta<1\right)$ are classes of starlike and convex functions of order $\beta,$ respectively,

• (4)

$\mathcal{S}^{\ast}\left(1,\left(\frac{1+z}{1-z}\right)^{\beta}\right)=\mathcal{S}_{\beta}^{\ast}$ and $\mathcal{C}\left(1,\left(\frac{1+z}{1-z}\right)^{\beta}\right)=\mathcal{C}_{\beta}$ are class of strongly starlike and convex functions of order $\beta,$ respectively,

(

• (5)

$\mathcal{S}^{\ast}\left(1,\sqrt{1+z}\right)=\mathcal{S}_{L}^{\ast}=\left\{f\in\mathcal{A}:\left|\left(\frac{zf^{\prime}(z)}{f(z)}\right)^{2}-1\right|<1\right\}$ is class of lemniscate starlike functions,

• (6)

$\mathcal{S}^{\ast}(\gamma,(1+z)\diagup(1-z))=\mathcal{S}^{\ast}[\gamma]$ and $\mathcal{C}(\gamma,(1+z)\diagup(1-z))=\mathcal{C}[\gamma]$ $\left(\gamma\in\mathbb{C}\diagdown\{0\}\right)$ are classes of starlike and convex functions of complex order, respectively,

• (7)

$\mathcal{S}^{\ast}\left(1,q_{k}(z)\right)=k-\mathcal{S}_{P}^{\ast}=\left\{f\in\mathcal{A}:\Re\left(\frac{zf^{\prime}(z)}{f(z)}\right)>k\left|\frac{zf^{\prime}(z)}{f(z)}-1\right|\right\}$ is class of $k-$parabolik starlike functions,

• (8)

$\mathcal{C}\left(1,q_{k}(z)\right)=k-\mathcal{UCV}=\left\{f\in\mathcal{A}:\Re\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)>k\left|\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right|\right\}$ is class of $k-$uniformly convex functions.

Here, for $0\leq k<\infty$ the function $q_{k}:$ $\mathbb{D\rightarrow\{}w=u+iv\in\mathbb{C}:$ $\ u^{2}>k^{2}\left((u-1)^{2}+v^{2}\right),u>0\mathbb{\}}$ has the form $q_{k}(z)=1+Q_{1}z+Q_{2}z^{2}+...,$ $\left({z\in}\mathbb{D}\right)$ where

[TABLE]

with $\mathcal{B}=\frac{2}{\pi}\arccos k$ and $\mathcal{K}(t)$ is the complete elliptic integral of first kind (see [20]).

A function $f\in\mathcal{A}$ is said to be bi-univalent in $\mathbb{D}$ if both $f$ and its inverse map $f^{-1}$ are univalent in $\mathbb{D}.$ Let $\sigma$ be the class functions $f\in\mathcal{S}$ that are bi-univalent in $\mathbb{D}$. For a brief history and interesting examples of functions which are in (or are not in) in the class $\sigma$, including various properties of such functions we refer the reader to the work of Srivastava et al. [5] and references therein. Bounds for the first few coefficients of various subclasses of bi-univalent functions were obtained by a variety of authors including [19, 13, 2], [1], [21], [6, 7, 17, 18]. Not much was known about the bounds of the general coefficients $a_{n};n\geq 4$ of subclasses of $\sigma$ up until the publication of the article [11] by Jahangiri and Hamidi and followed by a number of related publications. Moreover, many author have considered the Fekete-Szegö problem for various subclasses of $\mathcal{A}$, the upper bound for $\left|a_{3}-\mu a_{2}^{2}\right|$ is investigated by many different authors $\left(\text{see}\left[\text{\cite[cite]{[\@@bibref{}{HM3}{}{}]}},\text{\cite[cite]{[\@@bibref{}{zapr14}{}{}]}},\text{\cite[cite]{[\@@bibref{}{hoafr}{}{}]}},\text{\cite[cite]{[\@@bibref{}{hem12}{}{}]}}\right]\right).$ In this paper, we apply the Fekete-Szegö inequality to certain subclass of generalized bi-subordinate functions of complex order.

2. Coefficient Estimates

In the sequel, it is assumed that $\varphi$ is an analytic function with positive real part in the unit disk $\mathbb{D}$, satisfying $\varphi(0)=1,$ $\varphi^{\prime}(0)>0,$ and $\varphi(\mathbb{D})$ is symmetric with respect to the real axis. Such a function is known to be typically real with the series expansion $\varphi(z)=1+B_{1}z+B_{2}z^{2}+B_{3}z^{3}+...$ where $B_{1}$, $B_{2}$ are real and $B_{1}>0.$ Motivated by a class of functions defined by the first author [2], we define the following comprehensive class of analytic functions

[TABLE]

A function $f\in\mathcal{A}$ is siad the be generalized bi-subordinate of complex order $\gamma$ and type $\lambda$ if both $f$ and its inverse map $g=f^{-1}$ are in  $\mathcal{S}(\lambda,\gamma;\varphi)$. As special cases of the class  $\mathcal{S}(\lambda,\gamma;\varphi)$ we have  $\mathcal{S}(0,\gamma;\varphi)\equiv$ $\mathcal{S}^{\ast}\left(\gamma;\varphi\right)$ and $\mathcal{S}\left(1,\gamma;\varphi\right)\equiv$ $\mathcal{C}\left(\gamma;\varphi\right)$.

To prove our next theorems, we shall need the following well-known lemma (see [3]).

Lemma 2.1**.**

(see [3])Let the function $w\in\mathcal{B}_{0}$ be given by

[TABLE]

then for by every complex number $s,$

[TABLE]

In the following theorem, we consider fuctional $\left|a_{3}-\mu a_{2}^{2}\right|$ for $\gamma$ nonzero complex number and $\mu\in\mathbb{C}.$

Theorem 2.1**.**

Let $\gamma$ be a nonzero complex number, $\mu\in\mathbb{C}$  and $0\leq\lambda\leq 1.$ If both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{S}(\lambda,\gamma;\varphi),$ then we obtain,

[TABLE]

[TABLE]

and

[TABLE]

where $s=\frac{B_{2}}{B_{1}}-\frac{4B_{1}\gamma\left(1+2\lambda\right)}{\left(1+\lambda\right)^{2}},$ $t=\frac{B_{2}}{B_{1}}$ and $\mathcal{{L}=}\left|\frac{B_{2}}{B_{1}}+\left(1-\mu\right)\frac{4B_{1}\gamma\left(1+2\lambda\right)}{\left(1+\lambda\right)^{2}}\right|+\left|\frac{B_{2}}{B_{1}}\right|.$

Proof.

Let $f(z)\in\mathcal{S}(\lambda,\gamma;\varphi)$ and $g=f^{-1}$. Then there are two functions $u(z)=c_{1}z+c_{2}z^{2}+\cdot\cdot\cdot\left(z\in\mathbb{D}\right)$ and $v(w)=d_{1}w+d_{2}w^{2}+\cdot\cdot\cdot,$ such that

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Equating coefficients of both side of equations (2.4) and (2.5) yield

[TABLE]

so that, on account of (2.6) and (2.7)

[TABLE]

and

[TABLE]

Taking into account (2.8), (2.9), (2.10) and Lemma 2.1, we obtain

[TABLE]

and

[TABLE]

Thus, we have

[TABLE]

where $s=\frac{B_{2}}{B_{1}}-\frac{4B_{1}\gamma\left(1+2\lambda\right)}{\left(1+\lambda\right)^{2}}$ and $t=\frac{B_{2}}{B_{1}}.$

Furthermore,

[TABLE]

As a result of this, we obtain

[TABLE]

where $\mathcal{L}=\left|\frac{B_{2}}{B_{1}}+\left(1-\mu\right)\frac{4B_{1}\gamma\left(1+2\lambda\right)}{\left(1+\lambda\right)^{2}}\right|+\left|\frac{B_{2}}{B_{1}}\right|.$

Thus the proof is completed. ∎

We next consider the case,  when $\gamma$ and $\mu$ are real. Then we have:

Theorem 2.2**.**

Let $\gamma>0$ and if both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{S}(\lambda,\gamma;\varphi),$ then for $\mu\in\mathbb{R},$

• (1)

If $\left|B_{2}\right|\geq B_{1},$ we have

[TABLE]

• (2)

If $\left|B_{2}\right|<B_{1},$ we have

[TABLE]

where ${\normalsize\mathcal{F}=}\frac{\left(1+\lambda\right)^{2}\left(B_{1}-\left|B_{2}\right|\right)}{2\gamma B_{1}^{2}\left(1+2\lambda\right)}.$

For each $\mu$ there is a function $f\in\mathcal{S}(\lambda,\gamma;\varphi)$ such that equality holds.

Proof.

Using (2.12) and Lemma 2.1, we obtain

[TABLE]

Now, the proof will be presented in two cases by considering $B_{1}$, $B_{2}\in\mathbb{R}$ and$\ B_{1}>0.$

Firstly, we want to consider the case with $\left|B_{2}\right|\geq B_{1}.$ Several possible cases need to further analyze.

• Case 1

: If $\mu\leq 1$, using (2.13) and Lemma 2.1, we obtain

[TABLE]

• Case 2

:  If $\mu>1$, again using (2.13) and Lemma 2.1, we obtain

[TABLE]

Finally, we want to consider the case with $\left|B_{2}\right|<B_{1}.$ By a similar way, several possible cases need to further analyze.

• (i)

Let $\mu\leq 1-{\normalsize\mathcal{F}},$ using (2.13) and Lemma 2.1, we have

[TABLE]

• (ii)

Let $1-{\normalsize\mathcal{F}<}\mu\leq 1,$ using (2.13) and Lemma 2.1, we obtain

[TABLE]

• (iii)

Let $1<\mu<1+{\normalsize\mathcal{F}},$ using (2.13) and Lemma 2.1, we obtain

[TABLE]

• (iv)

Let $\mu\geq 1+{\normalsize\mathcal{F}},$ using (2.13) and Lemma 2.1, we have

[TABLE]

Thus the proof is completed. ∎

Finally, we consider the case, when $\gamma$ nonzero complex number and $\mu\in\mathbb{C}.$ Then we get:

Theorem 2.3**.**

Let $\gamma$ be a nonzero complex number and let both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{S}(\lambda,\gamma;\varphi).$ Then for $\mu\in\mathbb{R}$ we have

• (1)

If $\ \frac{\left(1+\left|\sin\theta\right|\right)\left|B_{2}\right|}{2B_{1}}\geq 1,$ we have

[TABLE]

• (2)

If $\ \frac{\left(1+\left|\sin\theta\right|\right)\left|B_{2}\right|}{2B_{1}}<1,$ we obtain

[TABLE]

*where $k_{1}=\frac{B_{2}\left(1+\lambda\right)^{2}e^{i\theta}}{4B_{1}^{2}\left|\gamma\right|\left(1+2\lambda\right)},$ $l_{1=}\frac{\left(\left|B_{2}\right|-2B_{1}\right)\left(1+\lambda\right)^{2}}{4B_{1}^{2}\left|\gamma\right|\left(1+2\lambda\right)},$ *\left|\gamma\right|=\gamma e^{i\theta}\and $\mathcal{N=}\frac{\left(1+\lambda\right)^{2}\left[\left|B_{2}\right|\left(1+\left|\sin\theta\right|\right)-2B_{1}\right]}{4B_{1}^{2}\left|\gamma\right|\left(1+2\lambda\right)}.$

For each $\mu$ there is a function in $\mathcal{S}(\lambda,\gamma;\varphi)$ such that the equality holds.

Proof.

Suppose $f\left(z\right)=z+\sum_{k=2}^{\infty}a_{k}z^{k}\in\mathcal{S}(\lambda,\gamma;\varphi),$ using (2.12) and Lemma 2.1, then we obtain

[TABLE]

Taking $\left|\gamma\right|=\gamma e^{i\theta}$, $k_{1}=\frac{B_{2}\left(1+\lambda\right)^{2}e^{i\theta}}{4B_{1}^{2}\left|\gamma\right|\left(1+2\lambda\right)}$ and $l_{1}=\frac{\left(\left|B_{2}\right|-2B_{1}\right)\left(1+\lambda\right)^{2}}{4B_{1}^{2}\left|\gamma\right|\left(1+2\lambda\right)}$, a direct calculation with (2.14) shows that

[TABLE]

Now, we will make some discussions for several different cases by considering $B_{1}$, $B_{2}\in\mathbb{R}$ and$\ B_{1}>0.$

Firstly, we want to consider the case with $\frac{2B_{1}}{\left|B_{2}\right|}-\left|\sin\theta\right|<1.$ Several possible cases need to further analyze.

• Case 1

: Let $\mu\leq 1-\Re\left(k_{1}\right)$. Then from (2.15) and Lemma 2.1, we obtain

[TABLE]

• Case 2

: Let $\mu>1-\Re\left(k_{1}\right)$, then from (2.15) and Lemma 2.1, we yield

[TABLE]

Finally, we want to consider the case with $\frac{2B_{1}}{\left|B_{2}\right|}-\left|\sin\theta\right|>1.$ By a similar approximation, several possible cases need to further analyze.

• (i)

Let  $\mu\leq 1-\Re\left(k_{1}\right)+\mathcal{N},$ using (2.15) and Lemma 2.1, we have

[TABLE]

• (ii)

Let $1-\Re\left(k_{1}\right)+\mathcal{N}{\normalsize<}\mu\leq 1-\Re\left(k_{1}\right),$ using (2.15) and Lemma 2.1, we obtain

[TABLE]

• (iii)

Let $1-\Re\left(k_{1}\right)<\mu<1-\Re\left(k_{1}\right)-\mathcal{N},$ using (2.15) and Lemma 2.1, we obtain

[TABLE]

• (iv)

Let $\mu\geq 1-\Re\left(k_{1}\right)-\mathcal{N},$ using (2.15) and Lemma 2.1, we have

[TABLE]

Thus the proof is completed. ∎

Corollary 2.4**.**

Let $\gamma=1$ and $\lambda=0.$ If both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{S}\left[A,B\right],$ then using Theorem (2.1), (2.2) and (2.3), we obtain

$\left(1\right)$*  For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{C},$*

[TABLE]

$\left(2\right)$*  For  $\gamma>0$ and $\mu\in\mathbb{R},$*

[TABLE]

$\left(3\right)$*   For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{R},$*

[TABLE]

where $B_{1}=A-B$, $B_{2}=-B\left(A-B\right)$, $-1\leq B<A\leq 1,$ $\psi_{1}\left(A,B,\theta\right)=\frac{\left|B\right|\left(1+\left|\sin\theta\right|-\cos\theta\right)-2}{4\left(A-B\right)}$ and $\psi_{2}\left(A,B,\theta\right)=\frac{\left|B\right|\left(1+\left|\sin\theta\right|+\cos\theta\right)-2}{4\left(A-B\right)}.$

Corollary 2.5**.**

Let $\gamma=1$ and $\lambda=1.$ If both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{C}\left[A,B\right],$ then using Theorem (2.1), (2.2) and (2.3), we have

$\left(1\right)$*  For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{C},$*

[TABLE]

$\left(2\right)$*  For  $\gamma>0$ and $\mu\in\mathbb{R},$*

[TABLE]

$\left(3\right)$*   For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{R},$*

[TABLE]

where $B_{1}=A-B$, $B_{2}=-B\left(A-B\right)$, $-1\leq B<A\leq 1,$ $\phi_{1}\left(A,B,\theta\right)=\frac{\left|B\right|\left(1+\left|\sin\theta\right|-\cos\theta\right)-2}{3\left(A-B\right)}$ and $\phi_{2}\left(A,B,\theta\right)=\frac{\left|B\right|\left(1+\left|\sin\theta\right|+\cos\theta\right)-2}{3\left(A-B\right)}.$

Corollary 2.6**.**

Let $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\lambda=0.$ If both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{S}^{\ast}[\gamma],$ then similarly, using Theorem (2.1), (2.2) and (2.3), we obtain $\left(i\right)$  For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{C},$

[TABLE]

$\left(ii\right)$*  For  $\gamma>0$ and $\mu\in\mathbb{R},$*

[TABLE]

$\left(iii\right)$*   For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{R},$*

[TABLE]

where $B_{1}=2$, $B_{2}=2$, $\psi_{1}\left(\gamma,\theta\right)=\frac{\left(\left|\sin\theta\right|-\cos\theta-1\right)}{8\left|\gamma\right|\text{\ }}$ and $\psi_{2}\left(\gamma,\theta\right)=\frac{\left(\left|\sin\theta\right|+\cos\theta-1\right)}{8\left|\gamma\right|\text{\ }}$.

Corollary 2.7**.**

Let \gamma\in\mathbb{C}\backslash\left\{0\right\}\and $\lambda=1.$ Let both functions $f$ of the form (1.1) and its inverse maps $g=f^{-1}$ are in $\mathcal{C}[\gamma].$ Then similarly, using Theorem (2.1), (2.2) and (2.3), we have

$\left(i\right)$*  For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{C},$*

[TABLE]

$\left(ii\right)$*  For  $\gamma>0$ and $\mu\in\mathbb{R},$*

[TABLE]

$\left(iii\right)$*   For  $\gamma\in\mathbb{C}\backslash\left\{0\right\}$ and $\mu\in\mathbb{R},$*

[TABLE]

where $B_{1}=2$, $B_{2}=2$, $\phi_{1}\left(\gamma,\theta\right)=\frac{\left(\left|\sin\theta\right|-\cos\theta-1\right)}{6\left|\gamma\right|\text{\ }}$ and $\phi_{2}\left(\gamma,\theta\right)=\frac{\left(\left|\sin\theta\right|+\cos\theta-1\right)}{6\left|\gamma\right|\text{\ }}$.

Acknowledgement 1**.**

The research of E. Deniz and M. Çağlar were supported by the Commission for the Scientific Research Projects of Kafkas University, project number 2016-FM-67.