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LOGARITHMIC COEFFICIENTS OF SOME CLOSE-TO-CONVEX FUNCTIONS

Part of: Geometric function theory

Published online by Cambridge University Press:  02 November 2016

MD FIROZ ALI  and
A. VASUDEVARAO
MD FIROZ ALI
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India email ali.firoz89@gmail.com
A. VASUDEVARAO*
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India email alluvasu@maths.iitkgp.ernet.in
*
✉alluvasu@maths.iitkgp.ernet.in
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Abstract

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The logarithmic coefficients $\unicode[STIX]{x1D6FE}_{n}$ of an analytic and univalent function $f$ in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ with the normalisation $f(0)=0=f^{\prime }(0)-1$ are defined by $\log (f(z)/z)=2\sum _{n=1}^{\infty }\unicode[STIX]{x1D6FE}_{n}z^{n}$. In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of $|\unicode[STIX]{x1D6FE}_{n}|$, $n=1,2,3$, for such functions $f$.

Keywords

analyticunivalentstarlikeconvexclose-to-convex functionslogarithmic coefficient

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 228 - 237
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was supported by the University Grants Commission through a UGC-SRF Fellowship. The second author was supported by SERB (DST).

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