On \(\alpha\)-convex functions of order \(\beta\)
type: Item
instance of: scholarly article
title: On \(\alpha\)-convex functions of order \(\beta\)
zbMATH Open document ID: 0895.30008
DOI: 10.1155/S0161171297001051
author: Seiichi Fukui
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1998-09-06T00:00:00Z
full work available at URL: https://eudml.org/doc/47948
review text: If \(f(z) = z + a_2 z^2+\dots\) is analytic in the unit disc \(U\) and \(f(z)f'(z)/z \neq 0\) in \(U\), let \(0 \leq \alpha \leq 1\) and define \(\Phi_\alpha(z) = \alpha (1 + zf'(z)/f''(z)) + (1 - \alpha) z f'(z)/f(z)\). The class of starlike functions of order \(\beta\) (\(0 \leq \beta \leq 1\)) is defined by \(\text{Re }\Phi_0(z) > \beta\), and the class of convex functions of order \(\beta\) by \(\text{Re }\Phi_1(z)> \beta\) for all \(z \in U\). This paper discusses and proves some class inclusion properties of the classes of \(\alpha\)-convex functions of order \(\beta\), type 1 defined by \(\text{Re }\Phi_\alpha(z) > \beta\) and of type 2 defined by \(\text{Re }\Phi_\alpha(z) < \beta\) in \(U\). The type 1 class with \(\beta = 0\) was introduced in 1969 by Mocanu. It should be noted that \(f''(z)\) is misprinted as \(f'(z)\) throughout the paper. Also, \(P\) and \(p\) seem to be used interchangeably.
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1127594
zbMATH Keywords: \(\alpha\)-convex functions
reviewed by: James A. Hummel
MaRDI profile type: Publication
OpenAlex ID: W2072188538
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description: scientific article; zbMATH DE number 1127594