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The Schwarzian norm estimates for Janowski convex functions

Part of: Geometric function theory

Published online by Cambridge University Press:  12 February 2024

Md Firoz Ali Open the ORCID record for Md Firoz Ali [Opens in a new window]  and
Sanjit Pal Open the ORCID record for Sanjit Pal [Opens in a new window]
Md Firoz Ali
Affiliation:
Department of Mathematics, National Institute of Technology Durgapur, Durgapur 713209, West Bengal, India (ali.firoz89@gmail.com; fali.maths@nitdgp.ac.in; palsanjit6@gmail.com)
Sanjit Pal
Affiliation:
Department of Mathematics, National Institute of Technology Durgapur, Durgapur 713209, West Bengal, India (ali.firoz89@gmail.com; fali.maths@nitdgp.ac.in; palsanjit6@gmail.com)
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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.

Keywords

analytic functionsunivalent functionsJanowski convex functionsSchwarzian norm

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 17
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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