The Schwarzian norm estimates for Janowski convex functions

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

Md Firoz Ali, Sanjit Pal

{"title":"The Schwarzian norm estimates for Janowski convex functions","authors":"Md Firoz Ali, Sanjit Pal","doi":"10.1017/s0013091524000014","DOIUrl":null,"url":null,"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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0013091524000014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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.

查看原文

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。

求助全文

约1分钟内获得全文求助全文