Differential subordinations and inequalities in the complex plane
type: Item
instance of: scholarly article
title: Differential subordinations and inequalities in the complex plane
zbMATH Open document ID: 0633.34005
DOI: 10.1016/0022-0396(87)90146-X
author: Sanford S. Miller
author: Petru T. Mocanu
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Let f and F be analytic in the unit disc U. The function f is subordinate to F, written \(f\prec F\) or f(z)\(\prec F(z)\), if F is univalent, \(f(0)=F(0)\) and f(U)\(\subset F(U)\). The authors deal with second order differential subordinations of the form \((1)\quad \psi (p(z),zp'(z),z^ 2p''(z);z)\prec h(z),\) where \(\psi\) : \({\mathbb{C}}^ 3\times U\to {\mathbb{C}}\). They generalize their previous results [see, Mich. Math. J. 28, 157-171 (1981; Zbl 0439.30015)] on the case (1). With help from this generalization they prove some new inequalities, for example: Theorem 6. If p is analytic in U with \(p(0)=0\), then \(| zp'(z)| +z^ 2p''(z)/p(z)| <1\) implies that \(| p(z)| <1;\) Theorem 7. If p is analytic in U with \(p(0)=1\), and if \(Re[2p(z)- zp''(z)/p'(z)-1]>0,\) then Re p(z)\(>0\).
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 4029976
zbMATH Keywords: second order differential subordinations
MaRDI profile type: Publication
cites work: Q4220704
cites work: Subordination by Convex Functions
cites work: Second order differential inequalities in the complex plane
cites work: Differential subordinations and univalent functions
cites work: Univalent solutions of Briot-Bouquet differential equations
description: scientific article; zbMATH DE number 4029976