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AN APPLICATION OF SCHUR’S ALGORITHM TO VARIABILITY REGIONS OF CERTAIN ANALYTIC FUNCTIONS II

Part of: Geometric function theory

Published online by Cambridge University Press:  02 December 2021

MD FIROZ ALI ,
VASUDEVARAO ALLU  and
HIROSHI YANAGIHARA
MD FIROZ ALI
Affiliation:
Department of Mathematics, NIT Durgapur, Mahatma Gandhi Avenue, Durgapur 713209, West Bengal, India e-mail: ali.firoz89@gmail.com; firoz.ali@maths.nitdgp.ac.in
VASUDEVARAO ALLU*
Affiliation:
Discipline of Mathematics, School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar, Khordha 752050, Odisha, India
HIROSHI YANAGIHARA
Affiliation:
Department of Applied Science, Faculty of Engineering, Yamaguchi University, Tokiwadai, Ube 755, Japan e-mail: hiroshi@yamaguchi-u.ac.jp
*
e-mail: avrao@iitbbs.ac.in
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Abstract

We extend our study of variability regions, Ali et al. [‘An application of Schur algorithm to variability regions of certain analytic functions–I’, Comput. Methods Funct. Theory, to appear] from convex domains to starlike domains. Let $\mathcal {CV}(\Omega )$ be the class of analytic functions f in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega }$ . As an application of the main result, we determine the variability region of $\log f'(z_0)$ when f ranges over $\mathcal {CV}(\Omega )$ . By choosing a particular $\Omega $ , we obtain the precise variability regions of $\log f'(z_0)$ for some well-known subclasses of analytic and univalent functions.

Keywords

analytic functionunivalent functionconvex functionstarlike functionBanach spacenormSchur algorithmsubordinationvariability region

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 468 - 481
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The second author thanks SERB-MATRICS for financial support.

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