Coefficient and Fekete-Szeg\"o problem estimates for certain subclass of analytic and bi-univalent functions

TL;DR

This paper investigates coefficient bounds and Fekete-Szeg"o problem estimates for a specific subclass of analytic and bi-univalent functions defined by a particular real part condition involving the function's derivative, expanding understanding of these function classes.

Contribution

It introduces a new subclass of analytic and bi-univalent functions based on a real part inequality and derives coefficient bounds and Fekete-Szeg"o estimates for this class.

Findings

Derived upper bounds for initial coefficients of the subclass.

Established Fekete-Szeg"o problem estimates for the subclass.

Introduced a new subclass of bi-univalent functions with specific geometric properties.

Abstract

In this paper, we obtain the Fekete-Szeg\"{o} problem for the $k$-th $(k\geq1)$ root transform of the analytic and normalized functions $f$ satisfying the condition \begin{equation*} 1+\frac{\alpha-\pi}{2 \sin \alpha}< {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1+\frac{\alpha}{2\sin \alpha} \quad (|z|<1), \end{equation*} where $\pi/2\leq \alpha<\pi$. Afterwards, by the above two-sided inequality we introduce and investigate a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szeg\"{o} problem for functions belonging to this analytic and bi-univalent function class.

Equations

$$1+\frac{\alpha -\pi }{2\sin \alpha }<\mathrm{Re}\left\{\frac{z{f}^{\prime }(z)}{f(z)}\right\}<1+\frac{\alpha }{2\sin \alpha }(|z|<1),$$

$$f(z)=z+\sum _{n=2}^{\infty }{a}_n{z}^n=z+a_2{z}^2+a_3{z}^3+\cdots ,$$

$$p(z)=1+p_{1}z+p_2{z}^2+\cdots +p_n{z}^n+\cdots ,$$

$$k(z):=\frac{z}{(1-z{)}^2}=z+2z^2+3z^3+\cdots +n{z}^n+\cdots (z\in \Delta ).$$

$$f^{-1}(f(z))=z(z\in \Delta )$$

$$f(f^{-1}(w))=w(|w|<r_0(f);r_0(f)\ge 1/4),$$

$$f^{-1}(w)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots =:g(w).$$

$$\frac{z}{1-z},-\log (1-z)\mathrm{and}\frac{1}{2}\log \left(\frac{1+z}{1-z}\right),$$

$$\frac{w}{1+w},\frac{\exp (w)-1}{\exp (w)}\mathrm{and}\frac{\exp (2w)-1}{\exp (2w)+1},$$

$$z-\frac{1}{2}{z}^2\mathrm{and}\frac{z}{1-z^2}.$$

$$f(z)\prec g(z)\mathrm{or}f\prec g,$$

$$w(0)=0\mathrm{and}|w(z)|<1(z\in \Delta ),$$

$$f(z)\prec g(z)\iff f(0)=g(0)\mathrm{and}f(\Delta )\subset g(\Delta ).$$

$$\mathrm{Re}\left\{\frac{z{f}^{\prime }(z)}{f(z)}\right\}>\gamma (z\in \Delta ).$$

$$1+\frac{\alpha -\pi }{2\sin \alpha }<\mathrm{Re}\left\{\frac{z{f}^{\prime }(z)}{f(z)}\right\}<1+\frac{\alpha }{2\sin \alpha }(z\in \Delta ),$$

$$\varphi (\alpha ):=1+\frac{\alpha -\pi }{2\sin \alpha }(\pi /2\le \alpha <\pi ).$$

$$2{\varphi }^{\prime }(\alpha )=[(\pi -\alpha )\cot \alpha +1]\csc \alpha (\pi /2\le \alpha <\pi ),$$

$$\underset{\alpha \to {\pi }^{-}}{\lim }\varphi (\alpha )=\frac{1}{2},$$

$$\left(\frac{z{f}^{\prime }(z)}{f(z)}-1\right)\prec {\mathcal{B}}_{\alpha }(z)(z\in \Delta ),$$

$${\mathcal{B}}_{\alpha }(z):=\frac{1}{2i\sin \alpha }\log \left(\frac{1+z{e}^{i\alpha }}{1+z{e}^{-i\alpha }}\right)(z\in \Delta ).$$

$${\mathcal{B}}_{\alpha }(z)=\sum _{n=1}^{\infty }{A}_n{z}^n(z\in \Delta ),$$

$$A_n:=\frac{(-1{)}^{(n-1)}\sin n\alpha }{n\sin \alpha }(n=1,2,\dots ).$$

$${\Omega }_{\alpha }:=\left\{\zeta \in \mathbb{C}:\frac{\alpha -\pi }{2\sin \alpha }<\mathrm{Re}\left\{\zeta \right\}<\frac{\alpha }{2\sin \alpha },\frac{\pi }{2}\le \alpha <\pi \right\}.$$

$$1+\frac{\alpha -\pi }{2\sin \alpha }<\mathrm{Re}\left\{\frac{z{f}^{\prime }(z)}{f(z)}\right\}<1+\frac{\alpha }{2\sin \alpha }(z\in \Delta )$$

$$1+\frac{\alpha -\pi }{2\sin \alpha }<\mathrm{Re}\left\{\frac{w{g}^{\prime }(w)}{g(w)}\right\}<1+\frac{\alpha }{2\sin \alpha }(w\in \Delta ),$$

$$\mathrm{Re}\left\{\frac{z{f}^{\prime }(z)}{f(z)}\right\}>\frac{1}{2}(z\in \Delta )$$

$$\mathrm{Re}\left\{\frac{w{g}^{\prime }(w)}{g(w)}\right\}>\frac{1}{2}(w\in \Delta ),$$

$$\left|p_{2}-\mu p_{1}^{2}\right|\leq\left\{\begin{array}[]{ll}-4\mu+2,&\hbox{if\ \ $\mu\leq 0$;}\\ 2,&\hbox{if\ \ $0\leq\mu\leq 1$;}\\ 4\mu-2,&\hbox{if\ \ $\mu\geq 1$.}\end{array}\right.$$

$$p(z)=\frac{1}{2}(1+\nu )\frac{1+z}{1-z}+\frac{1}{2}(1-\nu )\frac{1-z}{1+z}(0\le \nu \le 1),$$

$$\frac{1}{p(z)}=\frac{1}{2}(1+\nu )\frac{1+z}{1-z}+\frac{1}{2}(1-\nu )\frac{1-z}{1+z}(0\le \nu \le 1),$$

Full text

[TABLE]

where $\pi/2\leq\alpha<\pi$. Afterwards, by the above two-sided inequality we introduce and investigate a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szegö problem for functions belonging to this analytic and bi-univalent function class.

1. Introduction

Let $\mathcal{A}$ be the class of functions $f$ of the form

[TABLE]

which are analytic in the open unit disk $\Delta=\{z\in\mathbb{C}:|z|<1\}$ and normalized by the condition $f(0)=f^{\prime}(0)-1=0$. Also let $\mathcal{P}$ be the class of functions $p$ analytic in $\Delta$ of the form

[TABLE]

such that ${\rm Re}\{p(z)\}>0$ for all $z\in\Delta$. The subclass of all functions in $\mathcal{A}$ which are univalent (one-to-one) in $\Delta$ is denoted by $\mathcal{S}$. A well-known example for the class $\mathcal{S}$ is the Koebe function which has the following form

[TABLE]

It is known that the Koebe function maps the open unit disk $\Delta$ onto the entire plane minus the interval $(-\infty,-1/4]$. Also, the well-known Koebe One-Quarter Theorem states that the image of the open unit disk $\Delta$ under every function $f\in\mathcal{S}$ contains the disk $\{w:|w|<\frac{1}{4}\}$, see [6, Theorem 2.3]. Therefore, according to the above, every function $f$ in the class $\mathcal{S}$ has an inverse $f^{-1}$ which satisfies the following conditions:

[TABLE]

and

[TABLE]

where

[TABLE]

We say that a function $f\in\mathcal{A}$ is bi-univalent in $\Delta$ if, and only if, both $f$ and $f^{-1}$ are univalent in $\Delta$. We denote by $\Sigma$ the class of all bi-univalent functions in $\Delta$. The following functions

[TABLE]

with the corresponding inverse functions, respectively,

[TABLE]

belong to the class $\Sigma$. It is clear that the Koebe function is not a member of the class $\Sigma$, also the following functions

[TABLE]

The initial coefficients estimate of the class of bi-univalent functions $\Sigma$ is studied by Lewin in 1967 and he obtained the bound $1.51$ for the modulus of the second coefficient $|a_{2}|$, see [12]. Afterward, Brannan and Clunie conjectured that $|a_{2}|\leq\sqrt{2}$, see [3]. Finally, in 1969, Netanyahu [14] showed that $\max_{f\in\Sigma}|a_{2}|=4/3$. For the another coefficients $a_{n}$ $(n\geq 3)$ the sharp estimate is presumably still an open problem.

Let $f$ and $g$ be two analytic functions in $\Delta$. We say that a function $f$ is subordinate to $g$, written as

[TABLE]

if there exists a Schwarz function $w:\Delta\rightarrow\Delta$ with the following properties

[TABLE]

such that $f(z)=g(w(z))$ for all $z\in\Delta$. In particular, if $g\in\mathcal{S}$, then we have the following geometric equivalence relation

[TABLE]

A function $f\in\mathcal{A}$ is called starlike (with respect to [math]) if $tw\in f(\Delta)$ whenever $w\in f(\Delta)$ and $t\in[0,1]$. We denote by $\mathcal{S}^{*}$ the class of all starlike functions in $\Delta$. Also, we say that a function $f\in\mathcal{A}$ is starlike of order $\gamma$ ($0\leq\gamma<1$) if, and only if,

[TABLE]

The class of the starlike functions of order $\gamma$ in $\Delta$ is denoted by $\mathcal{S}^{*}(\gamma)$. As usual we put $\mathcal{S}^{*}(0)\equiv\mathcal{S}^{*}$.

We recall that a function $f\in\mathcal{A}$ belongs to the class $\mathcal{M}(\alpha)$ if $f$ satisfies the following two-sided inequality

[TABLE]

where $\pi/2\leq\alpha<\pi$. The class $\mathcal{M}(\alpha)$ was introduced by Kargar et al. in [8]. We define the function $\phi$ as follows

[TABLE]

Since

[TABLE]

therefore for each $\alpha\in[\pi/2,\pi)$ we see that $\phi^{\prime}(\alpha)\neq 0$. On the other hand, since $\phi(\pi/2)=1-\pi/4\approx 0.2146$ and

[TABLE]

thus the class $\mathcal{M}(\alpha)$ is a subclass of the starlike functions of order $\gamma$ where $0.2146\leq\gamma<0.5$. By this fact that $\mathcal{S}^{*}(\gamma)\subset\mathcal{S}$ for each $\gamma\in[0,1)$, thus we conclude that the members of the class $\mathcal{M}(\alpha)$ are univalent in $\Delta$.

Now, we recall the following result for the class $\mathcal{M}(\alpha)$, see [8, Lemma 1.1].

Lemma 1.1**.**

Let $f(z)\in\mathcal{A}$ and $\pi/2\leq\alpha<\pi$. Then $f\in\mathcal{M}(\alpha)$ if, and only if,

[TABLE]

where

[TABLE]

The function $\mathcal{B}_{\alpha}(z)$ is convex univalent and has the form

[TABLE]

where

[TABLE]

Also we have $\mathcal{B}_{\alpha}(\Delta)=\Omega_{\alpha}$ where

[TABLE]

Very recently Sun et al. (see [22]) and Kwon and Sim (see [11]) have studied the class $\mathcal{M}(\alpha)$. Sun et al. showed if the function $f$ of the form (1.1) belongs to the class $\mathcal{M}(\alpha)$, then $|a_{n}|\leq 1$ while the estimate is not sharp. Subsequently, Kwon and Sim obtained sharp estimates on the initial coefficients $a_{2}$, $a_{3}$, $a_{4}$ and $a_{5}$ of the functions $f$ belonging to the class $\mathcal{M}(\alpha)$. The coefficient estimate problem for each of the Taylor-Maclaurin coefficients $|a_{n}|$ $(n=6,7,\ldots)$ is still an open question. Also, the logarithmic coefficients of the function $f\in\mathcal{M}(\alpha)$ were estimated by Kargar, see [7].

It is interesting to mention this subject that Brannan and Taha [4] introduced certain subclass of the bi-univalent function class $\Sigma$, denoted by $\mathcal{S}^{*}_{\Sigma}(\gamma)$ similar to the class of the starlike functions of order $\gamma$ $(0\leq\gamma<1)$. For each function $f\in\mathcal{S}^{*}_{\Sigma}(\gamma)$ they found non-sharp estimates for the initial Taylor-Maclaurin coefficients. Recently, motivated by the Brannan and Taha’s work, many authors investigated the coefficient bounds for various subclasses of the bi-univalent function class $\Sigma$, see for instance [5, 15, 16, 17, 18, 19, 20, 21].

In this paper, motivated by the aforementioned works, we introduce and investigate a certain subclass of $\Sigma$ similar to the class $\mathcal{M}(\alpha)$ as follows.

Definition 1.1**.**

Let $\pi/2\leq\alpha<\pi$. A function $f\in\Sigma$ is said to be in the class $\mathcal{M}_{\Sigma}(\alpha)$, if the following inequalities hold:

[TABLE]

and

[TABLE]

where $g$ is defined by (1.2).

Remark 1.1*.*

Upon letting $\alpha\rightarrow\pi^{-}$ it is readily seen that a function $f\in\Sigma$ is in the class $\mathcal{M}_{\Sigma}(1/2)$ if the following inequalities are satisfied:

[TABLE]

and

[TABLE]

where $g$ is defined by (1.2).

The following lemma will be useful.

Lemma 1.2**.**

(see [13])* Let the function $p$ be of the form belongs to the class $\mathcal{P}$. Then for any complex number $\mu$ we have*

[TABLE]

The result is sharp for the cases $\mu<0$ or $\mu>1$ if and only if $p(z)=\frac{1+z}{1-z}$ or one of its rotations. If $0<\mu<1$, then the equality holds if and only if $p(z)=\frac{1+z^{2}}{1-z^{2}}$ or one of its rotations. For the case $\mu=0$, the equality holds if and only if

[TABLE]

or one of its rotations. If $\mu=1$, the equality holds if and only if

[TABLE]

or one of its rotations.

This paper is organized as follows. In Section 2 we derive the Fekete-Szegö coefficient functional associated with the $k$-th root transform for functions in the class $\mathcal{M}(\alpha)$. In Section 3 we propose to find the estimates on the Taylor-Maclaurin coefficients $|a_{2}|$, $|a_{3}|$ and Fekete-Szegö problem for functions in the class $\mathcal{M}_{\Sigma}(\alpha)$ which we introduced in Definition 1.1.

2. Fekete-Szegö problem for the class $\mathcal{M}(\alpha)$

Recently, many authors have obtained the Fekete-Szegö coefficient functional associated with the $k$-th root transform for certain subclasses of analytic functions, see for instance [2, 9, 10]. In this section, we investigate this problem for the class $\mathcal{M}(\alpha)$. At first, we recall that for a univalent function $f$ of the form (1.1), the $k$-th root transform is defined by

[TABLE]

For $f$ given by (1.1), we have

[TABLE]

Equating the coefficients of (2.1) and (2.2) yields

[TABLE]

Now we have the following.

Theorem 2.1**.**

Let $\pi/2\leq\alpha<\pi$ and $f\in\mathcal{M}(\alpha)$. If $F$ is the $k$-th $(k\geq 1)$ root transform of the function $f$ defined by (2.1), then for any complex number $\mu$ we have

[TABLE]

where $\delta_{1}:=(1-k(1+\cos\alpha))/2$, $\delta_{2}:=(1+k(1-\cos\alpha))/2$ and $b_{2k+1}$ and $b_{k+1}$ are defined by (2.3). The result is sharp.

Proof.

Let $\pi/2\leq\alpha<\pi$. If $f\in\mathcal{M}(\alpha)$, then by Lemma 1.1 and by definition of subordination, there exists a Schwarz function $w$ such that

[TABLE]

where $\mathcal{B}_{\alpha}$ is defined by (1.3). We define

[TABLE]

It is clear that $p(0)=1$ and $p\in\mathcal{P}$. Relationships (1.4) and (2.6) give us

[TABLE]

where $A_{1}=1$ and $A_{2}=-\cos\alpha$. If we equate the coefficients of $z$ and $z^{2}$ on both sides of (2.5), then we get

[TABLE]

and

[TABLE]

From (2.3), (2.7) and (2.8), we get

[TABLE]

and

[TABLE]

where $k\geq 1$. Therefore

[TABLE]

If we apply the Lemma 1.2 and letting

[TABLE]

then we get the desired inequality (2.4).

From now, we shall show that the result is sharp. For the sharpness of the first and third cases of (2.4), i.e. $\mu\leq\delta_{1}$ and $\mu\geq\delta_{2}$, respectively, consider the function

[TABLE]

or one of its rotations. It is easy to see that $f_{1}$ belongs to the class $\mathcal{M}(\alpha)$ and

[TABLE]

The last equation shows that these inequalities are sharp. For the sharpness of the second inequality, we consider the function

[TABLE]

A simple calculation gives that

[TABLE]

Therefore the equality in the second inequality (2.4) holds for the $k$-th root transform of the above function $f_{2}$. This completes the proof of Theorem 2.1. ∎

The problem of finding sharp upper bounds for the coefficient functional $|a_{3}-\mu a_{2}^{2}|$ for different subclasses of the normalized analytic function class $\mathcal{A}$ is known as the Fekete-Szegö problem. Therefore, if we let $k=1$ in the Theorem 2.1, then we get the Fekete-Szegö problem for the class $\mathcal{M}(\alpha)$ which we give in the following corollary.

Corollary 2.1**.**

Let $\alpha\in[\pi/2,\pi)$ and $f\in\mathcal{M}(\alpha)$. Then for any complex number $\mu$ we have

[TABLE]

The result is sharp.

Putting $\alpha=\pi/2$ in the Corollary 2.1 we get the following.

Corollary 2.2**.**

Let the function $f$ be given by (1.1) satisfies the inequality

[TABLE]

Then for any complex number $\mu\in\mathbb{C}$ we have the following sharp inequalities

[TABLE]

If we let $\alpha\rightarrow\pi^{-}$ in the Corollary 2.1, then we have:

Corollary 2.3**.**

If the function $f$ of the form (1.1) is starlike of order $1/2$, then for any complex number $\mu\in\mathbb{C}$ the following sharp inequalities hold true.

[TABLE]

From (2.7) and (2.8) and the first case of the Lemma 1.2 we get.

Corollary 2.4**.**

If a function $f\in\mathcal{A}$ of the form $\eqref{f}$ belongs to the class $\mathcal{M}(\alpha)$ $(\pi/2\leq\alpha<\pi)$, then the following sharp inequalities hold.

[TABLE]

3. Coefficient estimate and Fekete-Szegö problem for the class $\mathcal{M}_{\Sigma}(\alpha)$

In this section, motivated by the Zaprawa’s work (see [23]) we shall obtain the Fekete-Szegö problem for the class $\mathcal{M}_{\Sigma}(\alpha)$. Also, we obtain upper bounds for the first coefficients $|a_{2}|$ and $|a_{3}|$ of the function $f$ of the form (1.1) belonging to the class $\mathcal{M}_{\Sigma}(\alpha)$. The coefficient estimate problem for each of the coefficients $|a_{n}|$ $(n\geq 4)$ is an open question. Moreover, we apply the same technique as in [1].

Theorem 3.1**.**

Let the function $f$ given by (1.1) be in the class $\mathcal{M}_{\Sigma}(\alpha)$ and $\pi/2\leq\alpha<\pi$. Then

[TABLE]

and for any real number $\mu$ we have

[TABLE]

Proof.

Let $f\in\mathcal{M}_{\Sigma}(\alpha)$ be of the form (1.1) and $g=f^{-1}$ be given by (1.2). Then by Definition 1.1, Lemma 1.1 and definition of subordination there exist two Schwarz functions $u:\Delta\rightarrow\Delta$ and $v:\Delta\rightarrow\Delta$ with the properties $u(0)=0=v(0)$, $|u(z)|<1$ and $|v(z)|<1$ such that

[TABLE]

and

[TABLE]

where $\mathcal{B}_{\alpha}$ is defined by (1.3). Now we define the functions $k$ and $l$, respectively as follows

[TABLE]

and

[TABLE]

or equivalently

[TABLE]

and

[TABLE]

It is clear that the functions $k$ and $l$ belong to class $\mathcal{P}$ and $|k_{i}|\leq 2$ and $|l_{i}|\leq 2$ $(i=1,2,\ldots)$. From (1.4), (3.2)-(3.5), we have

[TABLE]

and

[TABLE]

where $A_{1}=1$ and $A_{2}=-\cos\alpha$. Thus, upon comparing the corresponding coefficients in (3.6) and (3.7), we obtain

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

From equations (3.8) and (3.10), we can easily see that

[TABLE]

and

[TABLE]

If we add (3.9) to (3.11), we get

[TABLE]

Substituting (3.8), (3.10) and (3.12) into (3.13), we obtain

[TABLE]

Now, (3.8) and (3.14) imply that

[TABLE]

Since $|k_{2}|\leq 2$ and $|l_{2}|\leq 2$, (3.15) implies that

[TABLE]

which proves the first assertion (3.1) of Theorem 3.1. Now, if we subtract (3.11) from (3.9) and use of (3.12), we get

[TABLE]

From (3.15) and (3.16) it follows that

[TABLE]

where

[TABLE]

Since $|k_{2}|\leq 2$ and $|l_{2}|\leq 2$, we conclude that

[TABLE]

This completes the proof. ∎

Taking $\mu=0$ in the above Theorem 3.1 we get.

Corollary 3.1**.**

Let $f$ of the form (1.1) be in the class $\mathcal{M}_{\Sigma}(\alpha)$. Then

[TABLE]

If we let $\alpha\rightarrow\pi^{-}$ in the Theorem 3.1, we get the following.

Corollary 3.2**.**

If the function $f$ of the form (1.1) belongs to the class $\mathcal{M}_{\Sigma}(1/2)$, then $|a_{2}|\leq 1$ and

[TABLE]

where $\mu$ is real.