On the coefficient of The $n^{th}$ Cesaro mean of order $\alpha$ of bi-univalent functions

TL;DR

This paper introduces a new subclass of bi-univalent functions in the unit disk and provides coefficient estimates for the second and third coefficients, advancing the understanding of their geometric properties.

Contribution

It defines a novel subclass of bi-univalent functions and derives new bounds for their initial coefficients, extending existing coefficient estimate theories.

Findings

Derived bounds for |a_2| and |a_3| coefficients

Introduced a new subclass of bi-univalent functions

Provided preliminary results related to the subclass

Abstract

The purpose of the present paper is to introduce a new subclasses of the function class of bi-univalent functions defined in the open unit disc. Furthermore, we obtain estimates on the coefficients $|a_{2}|$ and $|a_{3}|$ for functions of this class. Some results related to this work will be briefly indicated.

Equations

$$f(z)=z+\sum _{n=2}^{\infty }{a}_n{z}^n$$

$$g(w)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^2-5a_2{a}_3+a_4)w^4+....$$

$$\psi (z)=1+B_{1}z+B_2{z}^2+B_3{z}^3+....,(B_1>0).$$

$$u(z)=b_{1}z+\sum _{n=2}^{\infty }{b}_n{z}^n,v(z)=c_{1}z+\sum _{n=2}^{\infty }{c}_n{z}^n,(|z|<1).$$

$$|b_1|\le 1,|b_2|\le 1-|b_1{|}^2,|c_1|\le 1,|c_2|\le 1-|c_1{|}^2.$$

$$\psi (u(z))=1+B_1{b}_{1}z+(B_1{b}_2+B_2{b}_1^2)z^2+...,|z|<1$$

$$\psi (v(w))=1+B_1{c}_{1}w+(B_1{c}_2+B_2{c}_1^2)w^2+...,|w|<1.$$

$$|a_{2}|\leq\left|\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-2\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-2\\ k-2\end{array}}\right)}\end{array}\right|\frac{B_{1}\sqrt{B_{1}}}{\sqrt{[|3B^{2}_{1}-4B_{2}|+4B_{1}]}}$$

$$|a_{3}|\leq\left[\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-3\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-3\\ k-3\end{array}}\right)}\end{array}\right]\left[(1-\frac{4}{3B_{1}})\frac{B^{3}_{1}}{{[|3B^{2}_{1}-4B_{2}|+4B_{1}]}}+\frac{B_{1}}{3}\right].$$

$$[{\delta }_n^{\alpha }f(z){]}^{{}^{\prime }}=\psi (u(z)),[{\delta }_n^{\alpha }g(w){]}^{{}^{\prime }}=\psi (v(w)),$$

$$[{\delta }_n^{\alpha }f(z){]}^{{}^{\prime }}=1+2A_2{a}_{2}z+3A_3{a}_3{z}^3+...,$$

$$2A_2{a}_2=B_1{b}_1,$$

$$3A_3{a}_3=B_1{b}_2+B_2{b}_1^2,$$

$$-2A_2{a}_2=B_1{c}_1,$$

$$3[2A_2^2{a}_2^2-A_3{a}_3]=B_1{c}_2+B_2{c}_1^2.$$

$$b_1=-c_1.$$

$$A_2^2{a}_2^2[3B_1^2-8B_2]=B_1^3(b_2+c_2).$$

$$|A_2^2{a}_2^2[3B_1^2-8B_2]|\le 2B_1^3(1-|b_1{|}^2).$$

$$6A_3{a}_3=6A_2^2{a}_2^2+B_1(b_2-c_2).$$

$$\left|arg\left\{\frac{(1-\lambda ){\delta }_{{}_n}^{\alpha }f(z)+\lambda z[{\delta }_{{}_n}^{\alpha }f(z){]}^{\prime }}{z}\right\}\right|<\frac{\pi \alpha }{2},\alpha (0<\alpha \le 1,\lambda \ge 1,z\in U),$$

$$\left|arg\left\{\frac{(1-\lambda ){\delta }_{{}_n}^{\alpha }g(w)+\lambda w[{\delta }_{{}_n}^{\alpha }g(w){]}^{\prime }}{w}\right\}\right|<\frac{\pi \alpha }{2},\alpha (0<\alpha \le 1,\lambda \ge 1,w\in U),$$

$$|a_{2}|\leq 2\alpha\left|\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-2\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-2\\ k-2\end{array}}\right)}\end{array}\right|\frac{{1}}{\sqrt{4^{k}(1+\lambda)^{2}+\alpha[2.3^{k}(1+\lambda)-4^{k}(1+\lambda)^{2}]}},$$

$$|a_{3}|\leq\left[\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-3\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-3\\ k-3\end{array}}\right)}\end{array}\right]\left[\frac{2\alpha}{(1+2\lambda)}+\frac{4\alpha^{2}}{(1+\lambda)^{2}}\right].$$

$$\frac{(1-\lambda ){\delta }_{{}_n}^{\alpha }f(z)+\lambda z[{\delta }_{{}_n}^{\alpha }f(z){]}^{\prime }}{z}=[p(z){]}^{\alpha },$$

$$\frac{(1-\lambda ){\delta }_{{}_n}^{\alpha }g(w)+\lambda w[{\delta }_{{}_n}^{\alpha }g(w){]}^{\prime }}{w}=[q(w){]}^{\alpha },$$

$$p(z)=1+p_{1}z+p_2{z}^2+p_3{z}^3+...,$$

$$q(w)=1+p_{1}w+q_2{w}^2+q_3{w}^3+....$$

$$(1+\lambda )A_2{a}_2=\alpha p_1,$$

$$(1+2\lambda )A_3{a}_3=\frac{1}{2}[2\alpha p_2+\alpha (\alpha -1)p_1^2],$$

$$-(1+\lambda )A_2{a}_2=\alpha q_1,$$

Full text

1. Introduction

Let $A$ denote the class of the functions $f$ of the form

[TABLE]

which are analytic in the open unit disc $U=\{z\in C:|z|<1$} and satisfy the normalization condition $f(0)=f^{{}^{\prime}}(0)-1=0$. Let $S$ be the subclass of A consisting of functions of the form (1) which are also univalent in $U$.

A function $f\in A$ is said to be in the class of strongly bi-starlike functions of order $\alpha(0<\alpha\leq 1)$, denoted by $S^{*}_{\Sigma}(\alpha)$, if each of the following conditions is satisfied:

$\left|\arg\left\{\frac{zf^{{}^{\prime}}(z)}{f(z)}\right\}\right|<\frac{\alpha\pi}{2},(|z|<1,0<\alpha\leq 1),$

and

$\left|\arg\left\{\frac{zg^{{}^{\prime}}(w)}{g(w)}\right\}\right|<\frac{\alpha\pi}{2},(|w|<1,0<\alpha\leq 1),$

where $g$ is the extension of $f^{-1}$ to $U$ (for details see ( Brannan and Taha [1]). And is said to be in the class of strongly bi-convex functions of order $\alpha(0<\alpha\leq 1)$, denoted by $K_{\Sigma}(\alpha)$, if it satisfies the following inequality

$\left|\arg\left\{{1+\frac{zf^{{}^{\prime\prime}}(z)}{f^{{}^{\prime}}(z)}}\right\}\right|<\frac{\alpha\pi}{2},(|z|<1,0<\alpha\leq 1)$.

and

$\left|\arg\left\{{1+\frac{wg^{{}^{\prime\prime}}(w)}{g^{{}^{\prime}}(w)}}\right\}\right|<\frac{\alpha\pi}{2},(|w|<1,0<\alpha\leq 1)$.

Where $g$ is the extension of $f^{-1}$ to $U$ . Recall that the Koebe one-quarter theorem [2] ensures that the image of $D$ under every univalent function $f\in S$ contains a disk of radius $\frac{1}{4}$. Thus every univalent function $f$ has an inverse $f^{-1}$ satisfying $f^{-1}(f(z))=z$, $(z\in D)$ and

$f^{-1}(f(w))=w$, $(|w|<r_{0}f,r_{0}f\geq\frac{1}{4})$.

[TABLE]

In recent years, many authors discussed estimate on the coefficients $|a_{2}|$ and $|a_{3}|$ for subclasses of bi-univalent function (see for example [3], [4], [5], [6], [7], [8]).

Let $f:D\rightarrow C$ be an analytic function on $D$ having taylor expansion $f(z)=\Sigma^{\infty}_{n=1}a_{n}z^{n},z\in D,$ with $a_{n}\in C$, $a_{1}=1$ $,n=1,2,3,...\ .$ A function $f\in S$ is bi-univalent in $D$ if both $f$ and $f^{-1}$ are univalent in $D$.

The object of the present paper is to introduce a new subclasses of the function class $\Sigma$ and to find estimates on the coefficients $|a_{2}|$ and $|a_{3}|$ for new functions in these new subclasses of the function class $\Sigma$.

We say that $\delta_{n}^{\alpha}f(z)$ is The $n^{th}$ Cesaro mean of order $\alpha\geq 0$ of $f$ is defined by

$\delta_{n}^{\alpha}f(z)=z+\sum^{\infty}_{n=2}A_{n}a_{n}z^{n}$

where

A_{n}={\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-n\\ k-n\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-1\\ k-n\end{array}}\right)}\end{array}},\ a_{1}=1.

Let $D$ denote the open unit disk in $C$. It is well known that outer functions are zero-free on the unit disk. Outer functions, which play an important role in $H_{p}$ theory to find a suitable finite (polynomial) approximation for the outer infinite series $f$ so that the approximant reduces the zero-free property of $f$, arise in the characteristic equation which determines the stability of certain nonlinear systems of differential equations. Recall that an outer function is a function $f\in H_{p}$ of the form

$f(z)=e^{i\gamma}e^{\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+e^{it}z}{1-e^{it}z}\log\psi(t)dt}$

where $\psi(t)\geq 0$, $\log\psi(t)$ is in $L^{1}$ and $\psi(t)$ is in $L^{p}$. See [9] for the definitions and classical properties of outer functions. Since any function $f$ in $H^{1}$ which has $1/f$ in $H^{1}$ is an outer function, then typical examples of outer functions can be generated by functions of the form $\prod_{k=1}^{n}(1-e^{i\theta_{k}}z)^{\alpha_{k}}$ for $-1<\alpha_{k}<1$.

We observe for outer functions that the standard Taylor approximants do not, in general, retain the zero-free property of $f$. It was shown in [10] that the Taylor approximating polynomials to outer functions can vanish in the unit disk. By using convolution methods that the classical Cesaro means, retains the zero-free property of the derivatives of bounded convex functions in the unit disk. The classical Cesaro means play an important role in geometric function theory (see [11],[12]).

Lemma 1.1**.**

If $h\in p$ then $|c_{k}|<1,$ for each $k$, where $p$ is the family of all functions $h$ analytic in $U$ for which $\Re\{h(z)\}>0$, then

$h(z)=1+c_{1}z+c_{2}z^{2}+c_{3}z^{3}+...\ ,z\in U$.

2. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS $H_{\Sigma}(\psi)$

In the sequel, it is assumed that $\varphi$ . is an analytic function with positive real part in the unit disk $D$, satisfying $\psi(0)=1,\psi^{{}^{\prime}}(0)>0$, and $\psi(D)$ is symmetric with respect to the real axis. Such a function has a Taylor series of the form

[TABLE]

Suppose that $u(z)$ and $v(z)$ are analytic in the unit disk D with $u(0)=v(0)=0$, $|u(z)|<1,|v(z)|<1$, and suppose that

[TABLE]

It is well known that

[TABLE]

By a simple calculation, we have

[TABLE]

and

[TABLE]

Definition 2.1**.**

[13]** A function $f\in\Sigma$ is said to be in the class $H_{\Sigma}(\psi)$ if and only if

$f^{{}^{\prime}}(z)\prec\psi(z),g^{{}^{\prime}}(z)\prec\psi(w)$,

where $g(w)=f^{-1}(w)$.

Theorem 2.2**.**

If $f$ given by (1) is in the class $H_{\Sigma}(k,\psi)$, then

[TABLE]

and

[TABLE]

Proof.

Let $f\in H_{\Sigma}(k,\psi)$ and $g=f^{-1}$. Where $a_{1}=1$ . Then there are analytic functions $u,v:D\rightarrow D$ given by (4) such that

[TABLE]

since

[TABLE]

$[\delta_{n}^{\alpha}g(w)]^{{}^{\prime}}=1-2A_{2}a_{2}w+3[2A_{2}^{2}a^{2}_{2}-A_{3}a_{3}]w^{3}+...\ ,$

it follows from (6), (7), (10) and (11) that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From (12) and (14), we get

[TABLE]

By adding (15) to (13), further computations using (12) and (16) lead to

[TABLE]

Also, from (16) and (17), together with (5), we obtain

[TABLE]

From (12) and (18) we get

|a_{2}|\leq\left|\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-2\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-2\\ k-2\end{array}}\right)}\end{array}\right|\frac{B_{1}\sqrt{B_{1}}}{\sqrt{[|3B^{2}_{1}-4B_{2}|+4B_{1}]}}.

Which, in view of the well-known inequalities $|b_{2}|\leq 2$ and $|c_{2}|\leq 2$ for functions with positive real part, gives us the desired estimate on $|a_{2}|$ as asserted in (8). By subtracting (15) from (13), further computations using (12) and (16) lead to

[TABLE]

From (5), (12), (16) and (19), it follows that

$|a_{3}|=\frac{6A_{2}^{2}|a_{2}|^{2}+B_{1}(|b_{2}|+|c_{2}|)}{6A_{3}}$

$\leq\frac{6A_{2}^{2}|a_{2}|^{2}+B_{1}(1-|b_{1}|^{2})+(1-|c_{1}|^{2})}{6A_{3}}$

$\leq\frac{[1-\frac{4}{3B_{1}}]A_{2}^{2}|a_{2}|^{2}}{A_{3}}+\frac{B_{1}}{3A_{3}}$

|a_{3}|\leq\left[\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-3\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-3\\ k-3\end{array}}\right)}\end{array}\right]\left[(1-\frac{4}{3B_{1}})\frac{B^{3}_{1}}{{[|3B^{2}_{1}-4B_{2}|+4B_{1}]}}+\frac{B_{1}}{3}\right].

∎

3. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS $Q_{\Sigma}(\alpha,\mu,\lambda)$

Definition 3.1**.**

*A function $f(z)$ given by (1) is said to be in the class $Q_{\Sigma}(\alpha,\mu,\lambda)$ if the following conditions are satisfied: For $f\in\Sigma,$

[TABLE]

and

[TABLE]

where the function $g$ defined by (2).

Theorem 3.2**.**

Let the function $f(z)$ given by (1) be in the class $Q_{\Sigma}(\alpha,\mu,\lambda)$, $n\in N_{0},0\leq\beta<1,\lambda\geq 1$. Then

[TABLE]

and

[TABLE]

Proof.

From (20) and (21), we can write

[TABLE]

and

[TABLE]

where $p(z)$ and $q(w)$ in $P$ and $a_{1}=1$, and have the forms

[TABLE]

and

[TABLE]

Now, equating the coefficients in (24) and (25), we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From (28) and (30), we obtain

[TABLE]

and

[TABLE]

Now, from (29), (31) and (33), we obtain

$2(1+2\lambda)A_{2}^{2}a^{2}_{2}=\alpha(p_{2}+q_{2})+\frac{1}{2}[2\alpha(\alpha-1)(p_{1}^{2}+q_{1}^{2})]$

$=\alpha(p_{2}+q_{2})+\frac{\alpha(\alpha-1)}{2}.\frac{2(1+\lambda)^{2}A_{2}^{2}a^{2}_{2}}{\alpha^{2}}\ .$

Therefore we have

$a^{2}_{2}=\frac{\alpha^{2}(p_{2}+q_{2})}{[(1+\lambda)^{2}+\alpha[(1+2\lambda)]-\lambda^{2}]]A_{2}^{2}}\ .$

Applying Lemma 1.1 for the coefficients $p_{2}$ and $q_{2}$, we immediately have

|a_{2}|\leq 2\alpha\left|\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-2\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-2\\ k-2\end{array}}\right)}\end{array}\right|\frac{{1}}{\sqrt{4^{k}(1+\lambda)^{2}+\alpha[2.3^{k}(1+\lambda)-4^{k}(1+\lambda)^{2}]}}.

This gives the bound as asserted in (22).

Next, in order to find the bound on $|a_{3}|$, we subtract (29) from (31) and obtain

$2[(1+2\lambda)(A_{3}a_{3}-A_{2}^{2}a^{2}_{2})$

=$\frac{1}{2}(2\alpha(p_{2}-q_{2})+\alpha(\alpha-1)(p^{2}_{1}-q^{2}_{1})),$

$a_{3}=\frac{\alpha(p_{2}-q_{2})}{2(1+2\lambda)A_{3}}+\frac{\alpha^{2}(p^{2}_{1}+q^{2}_{1})}{2(1+\lambda)^{2}A_{3}}$,

a_{3}=\left[\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-3\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-3\\ k-3\end{array}}\right)}\end{array}\right]\left[\frac{\alpha(p_{2}-q_{2})}{2(1+2\lambda)}+\frac{\alpha^{2}(p^{2}_{1}+q^{2}_{1})}{2(1+\lambda)^{2}}\right],

Applying Lemma 1.1 for the coefficients $p_{2}$ and $q_{2}$, we immediately have

|a_{3}|\leq\left[\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-3\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-3\\ k-3\end{array}}\right)}\end{array}\right]\left[\frac{2\alpha}{(1+2\lambda)}+\frac{4\alpha^{2}}{(1+\lambda)^{2}}\right].

This completes the proof of Theorem 3.2. ∎

4. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS $H_{\Sigma}(\beta,\mu,\lambda)$

Definition 4.1**.**

A function $f(z)$ given by (1) is said to be in the class $H_{\Sigma}(\beta,\mu,\lambda)$ if the following conditions are satisfied: For $f\in\Sigma,$

[TABLE]

and

[TABLE]

where the function $g(z)$ defined by (2).

Theorem 4.2**.**

. Let $f(z)$ given by (1) be in the class $H_{\Sigma}(\beta,\mu,\lambda),0\leq\beta<1,\mu\geq 0,$ and $\lambda\geq 1.$ Then

[TABLE]

and

[TABLE]

Proof.

. It follows from (34) and (35) that there exists $p,q\in P$ such that

[TABLE]

and

[TABLE]

where $a_{1}=1$, and have the forms

[TABLE]

and

[TABLE]

respectively. Equating coefficients in (38) and (39) yields

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

From (42) and (44), we have

[TABLE]

and

[TABLE]

Also, from (43) and (45), we find that

[TABLE]

[TABLE]

$\Rightarrow$

[TABLE]

which is the bound on $|a_{2}|$ as given in (36).

Next, in order to find the bound on $|a_{3}|$ by subtracting (45) from (43), we obtain

$2A_{3}(1+2\lambda)a_{3}=$

$2(1+2\lambda)A_{2}^{2}a^{2}_{2}+(1-\beta)(p_{2}-q_{2})$

or, equivalently

$a_{3}=\frac{2(1+2\lambda)A_{2}^{2}a^{2}_{2}}{2A_{3}(1+2\lambda)}$ $+\frac{(1-\beta)(p_{2}-q_{2})}{2A_{3}(1+2\lambda)}.$

Upon substituting the value of $a_{2}^{2}$ from (47), we obtain

$a_{3}=A_{3}\left[\frac{(1-\beta)^{2}(p^{2}_{1}+q^{2}_{1})}{2(1+\lambda)^{2}}+\frac{(1-\beta)(p_{2}-q_{2})}{2(1+2\lambda)}\right]$.

Applying Lemma 1.1 for the coefficients $p_{1},p_{2},q_{1}$ and $q_{2}$ we obtain

$a_{3}=A_{3}\left[\frac{4(1-\beta)^{2}}{(1+\lambda)^{2}}+\frac{2(1-\beta)(p_{2}-q_{2})}{(1+2\lambda)}\right].$

|a_{3}|\leq\left[\begin{array}[]{c}\frac{\left({\begin{array}[]{c}k+\alpha-1\\ k-3\end{array}}\right)}{\left({\begin{array}[]{c}k+\alpha-3\\ k-3\end{array}}\right)}\end{array}\right]\left[\frac{4(1-\beta)^{2}}{(1+\lambda)^{2}}+\frac{2(1-\beta)}{(1+2\lambda)}\right].

which is the bound on $|a_{3}|$ as asserted in (37). ∎

Remark**.**

1. For all $\alpha\geq 0$, and $k=n$ in Theorems 2.2, we obtain the corresponding results due to Zhigang and Qiuqiu [14]

Remark**.**

2. For all $\alpha\geq 0$, and $k=n$ in Theorems 3.2 and 4.2, we obtain the corresponding results due to Frasin and Aouf [4].