Properties of certain analytic functions
type: Item
instance of: scholarly article
title: Properties of certain analytic functions
zbMATH Open document ID: 0692.30003
author: Hitoshi Saitoh
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: If \(p\in {\mathbb{N}}\), let A(p) denote the class of functions f analytic in the unit disc U with Taylor expansion \(f(z)=z^ p+\sum^{\infty}_{k=0}a_{p+k}z^{p+k}\). It is shown that for f in A(p), and \(0<\alpha <p!/(p-j)!\), the inequality \(Re(f^{(j)}(z)z^{j- p})>\alpha,\) (z\(\in U)\) implies \[ Re(f^{(j-1)}(z)z^{j-p- 1})>\frac{(p-j+1)!2\alpha +p!}{(p-j+1)!(2p-2j+3)}\quad (z\in U) \] for \(1\leq j\leq p\). Similar inequalities are deduced for \((1- \lambda)f(z)+\lambda zf'(z)\) when \(\lambda\geq 0\), and a number of corollaries are obtained.
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 4132501
zbMATH Keywords: real parts
zbMATH Keywords: inequality
reviewed by: C. N. Linden
MaRDI profile type: Publication
full work available at URL: pjaa.65.131
OpenAlex ID: W2088243520
cites work: Differential inequalities and Carathéodory functions
cites work: Certain subclasses of Bazilevich functions of type \(\alpha\)
cites work: Q3818526
DOI: 10.3792/PJAA.65.131
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description: scientific article; zbMATH DE number 4132501