Notes on the norm of pre-Schwarzian derivatives of certain analytic functions

TL;DR

This paper corrects previous proofs and establishes sharp bounds for the norm of pre-Schwarzian derivatives of specific analytic functions, enhancing understanding of their geometric properties.

Contribution

It provides correct proofs and sharp bounds for the pre-Schwarzian derivative norm, correcting earlier inaccuracies in the literature.

Findings

Corrected proofs for bounds on pre-Schwarzian derivatives

Established sharp bounds for specific classes of analytic functions

Clarified the geometric implications of these bounds

Abstract

In this paper, we obtain sharp bounds for the norm of pre--Schwarzian derivatives of certain analytic functions. Initially this problem was handled by H. Rahmatan, Sh. Najafzadeh and A. Ebadian [Stud Univ Babe\c{s}--Bolyai Math {\bf61}(2): 155--162, 2016]. We pointed out that the proofs by Rahmatan et al. are incorrect and present correct proofs.

Equations

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

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

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

$$\alpha <\mathrm{Re}\left\{{\left(\frac{z}{f(z)}\right)}^2{f}^{\prime }(z)\right\}<\beta (z\in \Delta ).$$

$$P_{\alpha ,\beta }(z)=1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }z}{1-z}\right)(z\in \Delta ),$$

$$\varphi :=\frac{2\pi (1-\alpha )}{\beta -\alpha },$$

$$\frac{z{f}^{\prime }(z)}{f(z)}\prec 1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }z}{1-z}\right)(z\in \Delta ),$$

$${\left(\frac{z}{f(z)}\right)}^2{f}^{\prime }(z)\prec 1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }z}{1-z}\right)(z\in \Delta ),$$

$$||f||\le \frac{2(\beta -\alpha )}{\pi }\left(1-e^{2\pi i\frac{1-\alpha }{\beta -\alpha }}\right).$$

$$||f||\le \frac{3(\beta -\alpha )}{\pi }\left(1-e^{2\pi i\frac{1-\alpha }{\beta -\alpha }}\right).$$

$$||f||\le \frac{2(\beta -\alpha )}{\pi }\sqrt{4{\sin }^2(\varphi /2)+2{\pi }^2}-\frac{4\sin (\varphi /2)}{\sqrt{4{\sin }^2(\varphi /2)+2{\pi }^2}},$$

$$\frac{z{f}^{\prime }(z)}{f(z)}\prec 1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }z}{1-z}\right)(z\in \Delta ).$$

$$\frac{z{f}^{\prime }(z)}{f(z)}=1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)(z\in \Delta ),$$

$$\log \left\{\frac{z{f}^{\prime }(z)}{f(z)}\right\}=\log \left\{1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right\}(z\in \Delta ),$$

$$\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}=\frac{(\beta -\alpha )i}{\pi }\left[\frac{1}{z}\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)+\frac{(1-e^{i\varphi })w^{\prime }(z)}{(1-w(z))(1-e^{i\varphi }w(z))\left(1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right)}\right].$$

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

$$\log z=\ln |z|+i\arg z(z\in \Delta \setminus \{0\},-\pi <\arg z\le \pi ).$$

$$|\log z|\le \sqrt{|z-1{|}^2+{\pi }^2},$$

$$|\log z|\le \sqrt{{\left|\frac{z-1}{z}\right|}^2+{\pi }^2}.$$

$$\left|\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right|$$

$$=\frac{\sqrt{|1-e^{i\varphi }{|}^2|w(z){|}^2+{\pi }^2|1-w(z){|}^2}}{|1-w(z)|}$$

$$\le \frac{\sqrt{4{\sin }^2(\varphi /2)|w(z){|}^2+{\pi }^2(1+|w(z){|}^2)}}{1-|w(z)|}$$

$$\le \frac{\sqrt{4{\sin }^2(\varphi /2)|z{|}^2+{\pi }^2(1+|z{|}^2)}}{1-|z|}$$

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

$$=\left|\frac{(\beta -\alpha )i}{\pi }\left[\frac{1}{z}\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)+\frac{(1-e^{i\varphi })w^{\prime }(z)}{(1-w(z))(1-e^{i\varphi }w(z))\left(1+\frac{(\beta -\alpha )i}{\pi }\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right)}\right]\right|$$

$$\le \frac{(\beta -\alpha )}{\pi }\left[\frac{1}{|z|}\left|\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right|+\frac{\left|1-e^{i\varphi }\right||w^{\prime }(z)|}{\left|1-w(z)\right|\left|1-e^{i\varphi }w(z)\right|\left(1-\frac{(\beta -\alpha )}{\pi }\left|\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right|\right)}\right]$$

$$\le \frac{(\beta -\alpha )}{\pi }\left[\frac{1}{|z|}\left\{\frac{\sqrt{4{\sin }^2(\varphi /2)|z{|}^2+{\pi }^2(1+|z{|}^2)}}{1-|z|}\right\}+\frac{2\sin (\varphi /2)}{1-|z|-\frac{(\beta -\alpha )}{\pi }\sqrt{4{\sin }^2(\varphi /2)|z{|}^2+{\pi }^2(1+|z{|}^2)}}.\frac{1+|z|}{1-|z{|}^2}\right].$$

$$||f||$$

$$\le \underset{z\in \Delta }{\sup }\left\{\frac{(\beta -\alpha )}{\pi }\left[\frac{1+|z|}{|z|}\sqrt{4{\sin }^2(\varphi /2)|z{|}^2+{\pi }^2(1+|z{|}^2)}+\frac{2\sin (\varphi /2)(1+|z|)}{1-|z|-\frac{(\beta -\alpha )}{\pi }\sqrt{4{\sin }^2(\varphi /2)|z{|}^2+{\pi }^2(1+|z{|}^2)}}\right]\right\}$$

$$=\frac{2(\beta -\alpha )}{\pi }\sqrt{4{\sin }^2(\varphi /2)+2{\pi }^2}-\frac{4\sin (\varphi /2)}{\sqrt{4{\sin }^2(\varphi /2)+2{\pi }^2}}$$

$$\left|\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right|$$

$$=\frac{\sqrt{|1-e^{i\varphi }{|}^2|w(z){|}^2+{\pi }^2|1-e^{i\varphi }w(z){|}^2}}{|1-e^{i\varphi }w(z)|}$$

$$\le \frac{\sqrt{4{\sin }^2(\varphi /2)|w(z){|}^2+{\pi }^2(1+|w(z){|}^2)}}{1-|w(z)|}(|e^{i\varphi }|=1)$$

$$\le \frac{\sqrt{4{\sin }^2(\varphi /2)|z{|}^2+{\pi }^2(1+|z{|}^2)}}{1-|z|}.$$

$$\left|\log \left(\frac{1-e^{i\varphi }w(z)}{1-w(z)}\right)\right|,$$

$$f_{\alpha ,\beta }(z)$$

Full text

1. Introduction

Let $\Delta$ be the open unit disc on the complex plane $\mathbb{C}$. Let $\mathcal{H}$ be the family of all analytic functions and $\mathcal{A}\subset\mathcal{H}$ be the family of all normalized functions in $\Delta$. We denote by $\mathcal{U}$ the class of all univalent functions in $\Delta$ and denote by $\mathcal{LU}\subset\mathcal{H}$ the class of all locally univalent functions in $\Delta$. For a $f\in\mathcal{LU}$, we consider the following norm

[TABLE]

where the quantity $f^{\prime\prime}/f$ is often referred to as pre–Schwarzian derivative of $f$ and in the theory of Teichmüller spaces is considered as element of complex Banach spaces. We remark that $||f||<\infty$ if, and only if, $f$ is uniformly locally univalent in $\Delta$. Note that, $||f||\leq 6$ if $f$ is univalent in $\Delta$ and, conversely, $f$ is univalent in $\Delta$ if $||f|\leq 1$. Both of these bounds are sharp, see [1]. For more geometric properties of the function $f$ relating the norm, see [2, 4, 7] and the references therein.

We say that a function $f$ is subordinate to $g$, written by $f(z)\prec g(z)$ or $f\prec g$ where $f$ and $g$ belonging to the class $\mathcal{A}$, if there exists a Schwarz function $w(z)$ is analytic in $\Delta$ with

[TABLE]

such that $f(z)=g(w(z))$ for all $z\in\Delta$.

In the sequel, we recall two definition which are certain subclasses of analytic and normalized functions $\mathcal{A}$. First, we say that a function $f\in\mathcal{A}$ belongs to the class $\mathcal{S}(\alpha,\beta)$ if it satisfies the following two–sided inequality

[TABLE]

where $0\leq\alpha<1$ and $\beta>1$. The class $\mathcal{S}(\alpha,\beta)$ was introduced by Kuroki and Owa (cf. [6]). Also, we say that a function $f\in\mathcal{A}$ belongs to the class $\mathcal{V}(\alpha,\beta)$ if

[TABLE]

The class $\mathcal{V}(\alpha,\beta)$ was first introduced by Kargar et al., see [5].

Since the convex univalent function

[TABLE]

where

[TABLE]

maps $\Delta$ onto the domain $\Omega=\{\omega:\alpha<{\rm Re}\{\omega\}<\beta\}$ conformally, thus we have.

Lemma 1.1**.**

([6, Lemma 1.3])* Let $0\leq\alpha<1$ and $\beta>1$. Then $f\in\mathcal{S}(\alpha,\beta)$ if, and only if,*

[TABLE]

where $\phi$ is defined in (1.2).

Lemma 1.2**.**

([5, Lemma 1.1])* Let $0\leq\alpha<1$ and $\beta>1$. Then $f\in\mathcal{V}(\alpha,\beta)$ if, and only if,*

[TABLE]

where $\phi$ is defined in (1.2).

Rahmatan et al. (see [8]) estimated the norm of pre–Schwarzian derivatives of the function $f$ where $f$ belong to the classes $\mathcal{S}(\alpha,\beta)$ and $\mathcal{V}(\alpha,\beta)$. Both estimates and proofs are incorrect. Indeed, the estimates of $||f||$ were wrongly proven by Rahmatan, Najafzadeh and Ebadian are in the following form:

Theorem A: For $0\leq\alpha<1<\beta$, if $f\in\mathcal{S}(\alpha,\beta)$, then

[TABLE]

Theorem B: For $0\leq\alpha<1<\beta$, if $f\in\mathcal{V}(\alpha,\beta)$, then

[TABLE]

First, note that both bounds are complex numbers.

In this paper we give the best estimate for $||f||$ when $f\in\mathcal{S}(\alpha,\beta)$ and disprove the Theorem B. However, we show that $||f||<\infty$ when $f\in\mathcal{V}(\alpha,\beta)$.

2. The Main Results

The first result of the paper is the following.

Theorem 2.1**.**

Let $0\leq\alpha<1$ and $\beta>1$. If a function $f$ belongs to the class $\mathcal{S}(\alpha,\beta)$, then

[TABLE]

where $\phi$ is defined in (1.2). The result is sharp.

Proof.

Let that $0\leq\alpha<1$, $\beta>1$ and $\phi$ be given by (1.2). If $f\in\mathcal{S}(\alpha,\beta)$, by Lemma 1.1, then we have

[TABLE]

The above subordination relation (2.2) implies that

[TABLE]

or equivalently

[TABLE]

where $w(z)$ is the Schwarz function. From (2.3), differentiating on both sides, after simplification, we obtain

[TABLE]

It is well–known that $|w(z)|\leq|z|$ (cf. [3]) and also by the Schwarz–Pick lemma, for a Schwarz function the following inequality

[TABLE]

holds. Also, we know that if $\log$ is the principal branch of the complex logarithm, then we have

[TABLE]

Therefore, by the above equation (2.6), it is well–known that if $|z|\geq 1$, then

[TABLE]

while for $0<|z|<1$, we have

[TABLE]

Thus, it is natural to distinguish the following cases.

Case 1: $\left|\frac{1-e^{i\phi}w(z)}{1-w(z)}\right|\geq 1$.

By (2.7), we have

[TABLE]

for all $z\in\Delta$. We note that the above inequality is well defined also for $z=0$. Thus from (2.4), (2.5) and (2), we get

[TABLE]

However, we obtain

[TABLE]

and concluding the inequality (2.1).

Case 2: $\left|\frac{1-e^{i\phi}w(z)}{1-w(z)}\right|<1$.

By (2.8), we have

[TABLE]

Since in the Cases 1 and 2 we have the equal estimates for

[TABLE]

therefore, in this case also, the desired result will be achieved. For the sharpness, consider the function $f_{\alpha,\beta}(z)$ as follows

[TABLE]

where $\phi$ is defined in (1.2), $0\leq\alpha<1$ and $\beta>1$. A simple calculation, gives us

[TABLE]

and thus $f_{\alpha,\beta}(z)\in\mathcal{S}(\alpha,\beta)$. With the same proof as above we get the desired result. Also, the result is sharp for a rotation of the function $f_{\alpha,\beta}(z)$ as follows:

[TABLE]

This is the end of proof. ∎

Remark 2.1*.*

In the Theorem B, the authors estimated $||f||$ when $f\in\mathcal{V}(\alpha,\beta)$. But in the proof of this theorem [8, p. 160], wrongly, they used from the following equation

[TABLE]

where $P_{\alpha,\beta}$ is defined in (1.1). This means that $f$, simultaneously, belonging to the class $\mathcal{S}(\alpha,\beta)$ and $\mathcal{V}(\alpha,\beta)$. Next, we show that the best estimate for $||f||$ when $f\in\mathcal{V}(\alpha,\beta)$ does not exists.

Theorem 2.2**.**

Let $0\leq\alpha<1$ and $\beta>1$. If a function $f$ belongs to the class $\mathcal{V}(\alpha,\beta)$, then $||f||<\infty$.

Proof.

Let $0\leq\alpha<1$, $\beta>1$ and $f\in\mathcal{V}(\alpha,\beta)$. Then by Lemma 1.2 and by use of definition of subordination, we have

[TABLE]

where $w$ is Schwarz function and $\phi$ is defined in (1.2). Taking logarithm on both sides of (2.10) and differentiating, we get

[TABLE]

With a simple calculation, (2.10) implies that

[TABLE]

Combining (2.11) and (2.12), give us

[TABLE]

It was proved in ([5, Theorem 2.2]) that if $f\in\mathcal{V}(\alpha,\beta)$ where $0<\alpha\leq 1/2$ and $\beta>1$, then

[TABLE]

Since ${\rm Re}\{z\}\leq|z|$, the last two–sided inequality means that $|f(z)/z|<\infty$ when $f\in\mathcal{V}(\alpha,\beta)$. Thus from the above we deduce that

[TABLE]

and concluding the proof. ∎