Briot--Bouquet differential superordinations and sandwich theorems
type: Item
instance of: scholarly article
title: Briot--Bouquet differential superordinations and sandwich theorems
zbMATH Open document ID: 1138.30009
author: Sanford S. Miller
author: Petru T. Mocanu
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-26T00:00:00Z
review text: Given complex numbers \(\beta ,\gamma\) and \(\Omega\subset\mathbb{C}\), an analytic function \(p:U\rightarrow\mathbb{C}\) in the unit disk is a solution of the Briot--Bouquet differential superordination, if \(\Omega\subset\{\widetilde{p}(z):=p(z)+zp'(z)(\beta p(z)+\gamma)^{-1} | z\in U\}\). In this article, properties are determined for functions \(p\) satisfying this differential superordination, as well as related sandwich theorems. To give an idea of the results obtained, let \(q:U\rightarrow\mathbb{C}\) be univalent with \(q(0)=a\). Considering now \(\widetilde{q}(z)\), suppose that \(\text{Re}(\beta q(z)+\gamma )>0\) and that \(\widetilde{q}(z)-q(z)\) is starlike. If \(p(z)=a+a_{1}z+a_{2}z^{2}+\cdots\) is analytic and injective in \(\overline{U}\setminus\{\zeta\in\partial U | \lim_{z\rightarrow\zeta}f(z)=\infty\}\), and \(\widetilde{p}\) is univalent in \(U\), then \(\widetilde{q}\prec\widetilde{p}\) implies that \(q\prec p\). Moreover, \(q\) is the best subordinant.
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 5129648
zbMATH Keywords: differential subordination
zbMATH Keywords: univalent
zbMATH Keywords: convex
zbMATH Keywords: starlike
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.05.080
OpenAlex ID: W2025647833
cites work: Q3147396
cites work: Q5476042
cites work: Subordination by Convex Functions
cites work: Univalent solutions of Briot-Bouquet differential equations
cites work: On some classes of first-order differential subordinations
cites work: Briot-bouquet differential equations and differential subordinations
cites work: Subordinants of differential superordinations
cites work: On normal and automorphic functions
DOI: 10.1016/J.JMAA.2006.05.080
description: scientific article; zbMATH DE number 5129648