On certain subclass of analytic and univalent functions in the unit disk
type: Item
instance of: scholarly article
title: On certain subclass of analytic and univalent functions in the unit disk
zbMATH Open document ID: 0532.30011
published in: Bulletin of the Iranian Mathematical Society
publication date: 1983-01-01T00:00:00Z
review text: The motivation for writing this paper seems to be two papers [Nieuw Arch. Wiskunde, III. Ser. 15, 55-63 (1967; Zbl 0172.375); and Indian J. Math. 13, 141-145 (1971; Zbl 0261.30011)] by the reviewer. Let S denote the class of functions \(f(z)=z+\sum^\infty_{n=2}a_ nz^ n\) analytic and univalent in the unit disc U. Let \(S_{\alpha}(\beta)\) denote the subclass of functions in S which satisfy the condition \[ | \frac{D^{\alpha}_ zf(z)}{z^{1-\alpha}}-\beta |<\beta,\quad z\in U,\quad 0<\alpha<1 \] and \(\beta>\frac{1}{2\Gamma(2-\alpha)}\) where \(D_ z^{\alpha}f(z)\) denotes the fractional derivative of order \(\alpha\) of f(z). By following the techniques used by the reviewer in the papers referred above, the author proves a distortion theorem and determines the coefficient estimates of functions in the class \(S_{\alpha}(\beta)\). It is further shown that the class \(S_{\alpha}(\beta)\) is closed under convolution and the functions in this class form a convex set.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 3844082
zbMATH Keywords: subclass of univalent functions
zbMATH Keywords: distortion theorems
zbMATH Keywords: convex set
author: Shigeyoshi Owa
reviewed by: R. M. Goel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3844082