Some inequalities involving meromorphically multivalent functions
type: Item
instance of: scholarly article
title: Some inequalities involving meromorphically multivalent functions
zbMATH Open document ID: 0887.30015
author: Oh Sang Kwon
author: Nak Eun Cho
author: Shigeyoshi Owa
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-05-12T00:00:00Z
review text: Let \(\Sigma_p\) be the class of functions of the form \(f(z) = z^{-p} + \sum_{k=-p+1}^\infty a_k z^k\) which are regular in the unit disc \(U\) except for the pole at \(z = 0\). In 1947, Robinson proved that if \(S(z)\) and \(T(z)\) are regular in \(U\) with \(\text{Re} \{zS'(z)/S(z)\} > 0\), \(|T'(z)/S'(z)|< 1\) for all all \( z \in U\), and \(T(0) = 0\), then \(|T(z)/S(z)|< 1\) for all \(z \in U\). Using variants of Jack's Lemma, the authors prove several theorems similar to Robinson's for functions in \(\Sigma_p\). The results are also motivated by earlier inequalities of Owa, Nunokawa, and Saitoh. The theorems proved are rather technical. For example, Theorem 2 of the paper proves that if \(S(z) \in \Sigma_m\), \(T(z) \in \Sigma_n\), \(p = n - m \geq 1\), \(S(z)\) satisfies \(\text{Re} \{S(z)/(zS'(z))\} < \;-\alpha\), \((0 \leq \alpha < 1/m)\), \(-\alpha / p \leq \roman {Im}\{-S(z)/(zS'(z))\} \leq \alpha/p\), and \(\text{Re} \{z^p T'(z)/S'(z)\} < 0\) for all \(z \in U\), then \(\text{Re} \{z^p T(z)/S(z)\} > 0\) for all \(z \in U\).
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1053990
reviewed by: James A. Hummel
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5507
OpenAlex ID: W2022694400
cites work: Functions Starlike and Convex of Order α
cites work: Second order differential inequalities in the complex plane
cites work: On properties of non-Carathéodory functions
cites work: Some inequalities involving multivalent functions
cites work: Univalent Majorants
cites work: Q4338920
DOI: 10.1006/JMAA.1997.5507
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description: scientific article; zbMATH DE number 1053990