Classes of uniformly convex and uniformly starlike functions as dual sets
type: Item
instance of: scholarly article
title: Classes of uniformly convex and uniformly starlike functions as dual sets
zbMATH Open document ID: 0888.30013
author: I. R. Nezhmetdinov
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: Let \(A\) denote the class of functions regular in the unit disc \(E\) which are of the form \(f(z)= z+\sum_{n=2}^\infty a_nz^n\) and let UCV and USV denote respectively the class of uniformly convex and uniformly starlike functions in \(E\). In the present paper, the author first shows that the classes UCV and USV can be expressed as dual sets, in the sense of convolution theory, of certain function families. He then uses these results to obtain some sharp sufficient conditions for functions in \(A\) to belong to the above classes. For instance, it is shown that if \(f\in A\) satisfies the condition \(\sum_{n=2}^\infty n(2n-1)|a_n|\leq 1\), then \(f\in \text{UCV}\), the constant 1 on the right hand side being the best possible.
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1115521
zbMATH Keywords: uniformly starlike functions
reviewed by: O. P. Juneja
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5640
OpenAlex ID: W2007024509
cites work: On uniformly starlike functions
cites work: On uniformly convex functions
cites work: Q4343763
cites work: Duality for Hadamard Products With Applications to Extremal Problems for Functions Regular in the Unit Disc
cites work: Q4863071
cites work: Univalent Functions and Nonanalytic Curves
DOI: 10.1006/JMAA.1997.5640
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description: scientific article; zbMATH DE number 1115521