A note on subordination
type: Item
instance of: scholarly article
title: A note on subordination
zbMATH Open document ID: 0599.30029
DOI: 10.1155/S0161171286000236
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46098
review text: Let A denote the class of functions of the form \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) which are analytic in the unit disc in \({\mathbb{C}}\). Let S be the subclass of A consisting of univalent functions. Then a function f of S is said to be starlike of order \(\alpha\) if and only if \(Re\{zf'(z)/f(z)\}>\alpha\) for some \(\alpha\) \((0\leq \alpha <1)\). Further a function f of S is said to be convex of order \(\alpha\) if and only if \(Re\{zf''(z)/f(z)\}>\alpha\) for some \(\alpha\) \((0\leq \alpha <1).\) The author uses a result of \textit{T. J. Suffridge} [Duke Math. J. 37, 775- 777 (1970; Zbl 0206.362)] to prove that if f is convex of order \(\alpha\) then the image of \(D_ r=\{z:\) \(| z| \leq r\}\) \((0<r<1)\) under f' is contained in the image of \(D_ r\) under \(e^{4(\alpha -1)/(1-z)}\). The author uses the same method to prove that if f is starlike of order \(\alpha\) then the image of \(D_ r\) under f(z)/z is contained in the image \(D_ r\) under \(e^{4(1-\alpha)/(1-z)}\). These results are sharp.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 3966602
zbMATH Keywords: starlike of order \(\alpha \)
zbMATH Keywords: convex of order \(\alpha \)
author: Shigeyoshi Owa
reviewed by: David J. Hallenbeck
MaRDI profile type: Publication
OpenAlex ID: W2118589155
description: scientific article; zbMATH DE number 3966602