On the class of univalent functions starlike with respect to \(N\)-symmetric points
type: Item
instance of: scholarly article
title: On the class of univalent functions starlike with respect to \(N\)-symmetric points
zbMATH Open document ID: 1005.30005
author: I. R. Nezhmetdinov
author: Saminathan Ponnusamy
published in: Hokkaido Mathematical Journal
publication date: 2002-09-17T00:00:00Z
review text: Let \(f(z)=z+a_2z^2+a_3z^3+\cdots\) be an univalent function in the unit disk \(U\) and let the class of univalent functions starlike with respect to \(N\)-symmetric points, \(SSP_N\) be described with \(\text{Re}\{zf'(z)/f_N(z)\}>0\), \(z\in U\), where \(f_N(z)=z+\sum^\infty_{m=1}a_{mN+1} z^{mN+1}\). In this paper authors give a structural formulae for functions in \(SSP_N\) and use it to obtain a sharp upper bound of \(|f'(z)|\) when \(N\) is even. This completes the distortion theorem for \(SSP_N\) since the rest is given by \textit{P. Singh} and \textit{R. Chand} [Indian. J. Pure Appl. Math. 10, 1167--1174 (1979; Zbl 0416.30010)]. The authors also give a characterization of the class \(SSP_N\) in terms of Hadamard convolution and prove that for any \(N\geq 2\) the class \(SSP_N\) contains non-starlike functions motivated by the fact that \(SSP_1\equiv S^*\).
reviewed by: Nikola Tuneski
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1751372
zbMATH Keywords: univalent
zbMATH Keywords: starlike
zbMATH Keywords: close-to-convex and convex functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350911770
OpenAlex ID: W1993678900
DOI: 10.14492/HOKMJ/1350911770
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description: scientific article; zbMATH DE number 1751372