Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$

TL;DR

This paper corrects previous inaccuracies in estimating the norm of the pre-Schwarzian derivative for bi-starlike functions of order alpha, providing accurate proofs and bounds.

Contribution

It offers corrected proofs and bounds for the norm of the pre-Schwarzian derivative on bi-starlike functions of order alpha, improving upon prior work.

Findings

Corrected bounds for the pre-Schwarzian norm

Valid proofs for bi-starlike functions of order alpha

Clarification of previous inaccuracies

Abstract

In the present paper we estimate the norm of the pre-Schwarzian derivative of bi-starlike functions of order $\alpha$ where $\alpha\in[0,1)$. Initially this problem was handled by Rahmatan et al. in [Bull Iran Math Soc {\bf43}: 1037-1043, 2017]. We pointed out that the proofs and bounds by Rahmatan et al. are incorrect and present correct proofs and bounds.

Equations

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

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

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

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

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

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

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

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

$$f_1(z)=\frac{z}{1-z},f_2(z)=-\log (1-z)\mathrm{and}{f}_3(z)=\frac{1}{2}\log \left(\frac{1+z}{1-z}\right),$$

$$f_1^{-1}(w)=\frac{w}{1+w},f_2^{-1}(w)=\frac{\exp (w)-1}{\exp (w)}\mathrm{and}{f}_3^{-1}(w)=\frac{\exp (2w)-1}{\exp (2w)+1},$$

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

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

$$\mathrm{Re}\left\{{\left(\frac{z}{f(z)}\right)}^2{f}^{\prime }(z)\right\}>\alpha (0\le \alpha <1,z\in \Delta ).$$

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

$$T_f(z):=\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\mathrm{and}{S}_f(z):=T_f^{\prime }(z)-\frac{1}{2}{T}_f^2(z),$$

$$||f||=\underset{z\in \Delta }{\sup }(1-|z{|}^2)\left|\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right|.$$

$$||f||\le \min \{6-4\alpha ,4\alpha +2\}.$$

$$||f||\le \min \{10-8\alpha ,6-8\alpha \}.$$

$$||f||\leq\left\{\begin{array}[]{ll}6,&\hbox{$\alpha=0$;}\\ \min\left\{6-4\alpha,\frac{4(1-\alpha)}{\alpha}\right\},&\hbox{$0<\alpha<\frac{1}{2}$;}\\ 4,&\hbox{$\alpha=1/2$;}\\ 2,&\hbox{$1/2<\alpha<1$.}\end{array}\right.$$

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

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

$$F(w):=\frac{1+(1-2\alpha )w}{1-w}(w\in \Delta )$$

$$\frac{w{g}^{\prime }(w)}{g(w)}\prec F(w)(w\in \Delta ).$$

$$\frac{f(z)}{z{f}^{\prime }(z)}\prec \frac{1+(1-2\alpha )z}{1-z}(z\in \Delta ).$$

$$\frac{f(z)}{z{f}^{\prime }(z)}=\frac{1+(1-2\alpha )\varphi (z)}{1-\varphi (z)}(z\in \Delta ),$$

$$\frac{f^{\prime }(z)}{f(z)}=\frac{1-\varphi (z)}{z(1+(1-2\alpha )\varphi (z))}(0\ne z\in \Delta ).$$

$$\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}=\frac{2(\alpha -1)\varphi (z)}{z(1+(1-2\alpha )\varphi (z))}+\frac{2(\alpha -1){\varphi }^{\prime }(z)}{(1-\varphi (z))(1+(1-2\alpha )\varphi (z))}(0\ne z\in \Delta ).$$

$$|{\varphi }^{\prime }(z)|\le \frac{1-|\varphi (z){|}^2}{1-|z{|}^2}(z\in \Delta ).$$

$$\left|\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right|$$

$$\le \frac{2|\alpha -1||\varphi (z)|}{|z|(1-|1-2\alpha ||\varphi (z)|)}+\frac{2|\alpha -1||{\varphi }^{\prime }(z)|}{(1-|\varphi (z)|)(1-|1-2\alpha ||\varphi (z)|)}$$

$$\le \frac{2(1-\alpha )}{1-|1-2\alpha ||z|}+\frac{2(1-\alpha )}{1-|1-2\alpha ||z|}.\frac{1+|z|}{1-|z{|}^2}.$$

Full text

1. Introduction

Let $\mathcal{H}$ be the family of analytic functions $f$ on the open unit disc $\Delta=\{z\in\mathbb{C}:|z|<1\}$ and $\mathcal{A}$ be a subclass of $\mathcal{H}$ that its members are normalized by the condition $f(0)=0=f^{\prime}(0)-1$. This means that each $f\in\mathcal{A}$ has the following form

[TABLE]

We denote by $\mathcal{U}\subset\mathcal{A}$ the family of univalent (one–to–one) functions $f$ in $\Delta$. Let $f$ and $g$ belong to the class $\mathcal{H}$. A function $f$ is called to be subordinate to $g$, written as

[TABLE]

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

[TABLE]

such that $f(z)=g(\phi(z))$ for all $z\in\Delta$. If $g\in\mathcal{U}$, then the following geometric equivalence relation holds true:

[TABLE]

It is known that the Koebe function

[TABLE]

maps the open unit disc $\Delta$ 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 disc $\Delta$ under every function $f\in\mathcal{U}$ contains the disc $\{w:|w|<\frac{1}{4}\}$, see [3, Theorem 2.3]. Therefore every function $f$ in the class $\mathcal{U}$ has an inverse $f^{-1}$ which satisfies the following conditions:

[TABLE]

and

[TABLE]

where

[TABLE]

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 bi-univalent functions in $\Delta$. The functions

[TABLE]

with the corresponding inverse functions, respectively

[TABLE]

belong to the class $\Sigma$.

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 the starlike functions in $\Delta$. Also, we say that a function $f\in\mathcal{A}$ is starlike of order $\alpha$ ($0\leq\alpha<1$) if, and only if,

[TABLE]

The class of the starlike functions of order $\alpha$ in $\Delta$ is denoted by $\mathcal{S}^{*}(\alpha)$. Now by definition of the starlike functions of order $\alpha$ we recall the following definition.

Definition 1.1**.**

Let $\alpha\in[0,1)$. A function $f\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{S}^{*}_{\Sigma}(\alpha)$ if the conditions (1.3) and

[TABLE]

hold true where $w=f(z)$ and $g=f^{-1}$.

Denote by $\mathcal{V}(\alpha)$ the class of all $f\in\mathcal{A}$ satisfying the following condition

[TABLE]

The class $\mathcal{V}(\alpha)$ was considered by Obradović and Ponnusamy in [5]. Now we recall the following definition.

Definition 1.2**.**

Let $\alpha\in[0,1)$. A function $f\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{V}_{\Sigma}(\alpha)$ if the conditions (1.5) and

[TABLE]

hold true where $w=f(z)$ and $g=f^{-1}$.

Denote by $\mathcal{LU}$ the subclass of $\mathcal{H}$ consisting of all locally univalent functions, namely, $\mathcal{LU}:=\{f\in\mathcal{H}:f^{\prime}(z)\neq 0,\ \ z\in\Delta\}$. For a $f\in\mathcal{LU}$ the pre–Schwarzian and Schwarzian derivatives of $f$ are defined as follows

[TABLE]

respectively. We note that the quantity $T_{f}$ (resp. $S_{f}$) is analytic on $\Delta$ precisely when $f$ is analytic (resp. meromorphic) and locally univalent on $\Delta$. Since $\mathcal{LU}$ is a vector space over $\mathbb{C}$ (see [4]), thus we can define the norm of $f\in\mathcal{LU}$ by

[TABLE]

This norm has significance in the theory of Teichmüller spaces, see [1]. It is known that $||f||<\infty$ if and only if $f$ is uniformly locally univalent. Also, notice that if $||f||\leq 1$, then $f$ is univalent in $\Delta$ and conversely, if $f$ univalent in $\Delta$, then $||f||\leq 6$ and the equality is attained for the Koebe function and its rotation. In fact, both of these bounds are sharp, see [2]. Also, if $f$ is starlike of order $\alpha\in[0,1)$, then we have the sharp estimate $||f||\leq 6-4\alpha$ (see e.g. [7]).

The following theorems were wrongly proven by Rahmatan et al. [6, Theorem 2.2 and Theorem 2.4].

Theorem A. Let the function $f(z)$ given by (1.1) be in the class $\mathcal{S}^{*}_{\Sigma}(\alpha)$ where $\alpha\in[0,1)$. Then

[TABLE]

Theorem B. Let the function $f(z)$ given by (1.1) be in the class $\mathcal{V}_{\Sigma}(\alpha)$ where $\alpha\in[0,1)$. Then

[TABLE]

We notice that $6-8\alpha\in(-2,6]$ when $\alpha\in[0,1)$.

In the present paper we give a correct proof for Theorem A and some remarks on the Theorem B.

2. On the Theorem A

The correct version of Theorem A is contained in the following theorem:

Theorem 2.1**.**

Let $\alpha\in[0,1)$. If the function $f$ of the form (1.1) is in the class $\mathcal{S}^{*}_{\Sigma}(\alpha)$, then

[TABLE]

Proof.

Let $f\in\mathcal{S}^{*}_{\Sigma}(\alpha)$ be given by (1.1). Then by Definition 1.1 the following conditions hold true:

[TABLE]

and

[TABLE]

where $w=f(z)$ and $g=f^{-1}$. Since (2.2) holds true by [7] we conclude that $||f||\leq 6-4\alpha$ where $\alpha\in[0,1)$. On the other hand, since the mapping

[TABLE]

maps $\Delta$ onto the right half-plane having real part greater than $\alpha$, thus (2.3) implies that

[TABLE]

By the identity $\frac{{\rm d}}{{\rm d}w}f^{-1}(w)=\frac{1}{f^{\prime}(z)}$ (see [6, Eq. (1.2)]) (2.5) is equivalent to

[TABLE]

Now by (2.6) and definition of subordination there exists a Schwarz function $\phi$ with $\phi(0)=0$ and $|\phi(z)|<1$ $(z\in\Delta)$ such that

[TABLE]

or equivalently

[TABLE]

Taking logarithm on both sides of (2.7) and differentiating, after simplification, we obtain

[TABLE]

The well-known Schwarz-Pick lemma states that for a Schwarz function $\phi$ the following inequality holds true:

[TABLE]

Now by (2.9) and since $|\phi(z)|\leq|z|$ (see [3]), the relation (2.8) implies that

[TABLE]

However we get

[TABLE]

Now we consider the following cases.

Case 1. $\alpha=0,1/2$. In this case we have $\varphi(\alpha)=\infty$. Therefore

[TABLE]

Case 2. $0<\alpha<1/2$. In this case $\varphi(\alpha)$ becomes $\frac{2(1-\alpha)}{\alpha}\in(2,\infty)$. Also we have $6-4\alpha\in(2,4)$ when $\alpha\in(1/2,1)$. Thus

[TABLE]

Case 3. $1/2<\alpha<1$. In this case we have $\varphi(\alpha)=2$. Thus

[TABLE]

Because $2<6-4\alpha$ when $\alpha\in(1/2,1)$. This completes the proof. ∎

3. On the Theorem B

In this section we give some remarks on the Theorem B. Let a function $f$ be of the form (1.1) belongs to the class $\mathcal{V}_{\Sigma}(\alpha)$. Then the conditions (1.5) and (1.6) hold true. From the geometric meaning of the function $(1+(1-2\alpha)z)/(1-z)$, subordination principle and (1.5) we have

[TABLE]

Using definition of subordination there exists a Schwarz function $\phi$ so that

[TABLE]

With a simple calculation (3.1) yields

[TABLE]

Again, from (3.1) we get

[TABLE]

By substituting (3.3) into (3.2), we have

[TABLE]

To estimate the above relationship (3.4), we need to estimate $|f(z)/z|$, but so far this is an open question. So this question seems to be more difficult than what is given in [6]. Also, we remark that in [6, proof of Theorem 2.4] Rahmatan et al. mistakenly used the following relation

[TABLE]

and this means that $f\in\mathcal{S}^{*}(\alpha)$. Thus Theorem 2.4 and its proof in [6] is incorrect.