The Schwarzian norm estimates for Janowski convex functions

Pub Date : 2024-02-12 DOI:10.1017/s0013091524000014

Md Firoz Ali, Sanjit Pal

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引用次数: 0

Abstract

For

$-1\leq B \lt A\leq 1$

, let

$\mathcal{C}(A,B)$

denote the class of normalized Janowski convex functions defined in the unit disk

$\mathbb{D}:=\{z\in\mathbb{C}:|z| \lt 1\}$

that satisfy the subordination relation

$1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$

. In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class

$\mathcal{C}(A,B)$

. The Dieudonné’s lemma which gives the exact region of variability for derivatives at a point of bounded functions, plays the key role in this study, and we also use this lemma to construct the extremal functions for the sharpness by a new method.

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扬诺夫斯基凸函数的施瓦兹规范估计值

对于$-1/leq B \lt A\leq 1$，让$\mathcal{C}(A,B)$表示定义在单位盘$\mathbb{D}:=\{z\in\mathbb{C}:|z|\lt 1\}$中满足从属关系$1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$ 的归一化扬诺夫斯基凸函数类。在本文中，我们确定了类$\mathcal{C}(A,B)$ 中函数的施瓦兹规范的尖锐估计值。Dieudonné Lemma 给出了有界函数在某一点上导数的精确可变区域，它在本研究中发挥了关键作用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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