Differential subordination of a harmonic mean to a linear function
type: Item
instance of: scholarly article
title: Differential subordination of a harmonic mean to a linear function
zbMATH Open document ID: 1426.30008
DOI: 10.1216/RMJ-2018-48-5-1475
author: Oliwia Chojnacka
author: Adam Lecko
published in: Rocky Mountain Journal of Mathematics
publication date: 2018-10-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1539936032
review text: The much studied subject of subordination of a function \(f\), analytic in the unit disc, to a function \(g\), univalent in the unit disc, defined via a composition of \(g\) with a normalised member of the class \(B\) of bounded analytic functions, has been extended to the concept of differential subordination [\textit{S. S. Miller} and \textit{P. T. Mocanu}, Differential subordinations: theory and applications. New York, NY: Marcel Dekker (2000; Zbl 0954.34003)]. Miller and Mocanu [loc. cit.] discuss the concept of differential subordination as a complex analogue of the concept of differential inequality aboundantly used in real analysis. The prime examples of the functions for which a differential inequality provides information about the range of the function in the complex case are furnished by the class \(P\) of analytic functions with positive real part as well as convex, star-like and close-to-convex classes. The focus of the paper under review is differential subordination to a linear dominant, plus some study of selected convex dominants. The particular brand of differential subordination studied is related to the harmonic mean. For two complex numbers \(a\) and \(b\), and a number \(t\) from the interval \([0,1]\), the harmonic mean of \(a\) and \(b\) is the ratio \(ab/[b+t(a-b)]\), provided the denominator is not zero. For any nonconstant function \(p\), defined on the unit disc, and a fixed \(t\) from the interval \((0,1)\), the families of functions of the type \[ [p(z)p(z)+zp'(z)]/[p(z)+(1-t)zp'(z)] \] which are either analytic in the unit disc or possess only removable singularities there and analytic extentions to the unit disc, are studied in the context of subordination to a linear function. The authors of the paper under review work with a technical lemma from [loc. cit.] and use it to prove a theorem that is a generalization of their own result from [Bull. Soc. Sci. Lett. Łódź, Sér. Rech. Déform. 64, No. 2, 29--40 (2014; Zbl 1320.30046)]. In the …
reviewed by: K. T. Hallenbeck
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 6958788
zbMATH Keywords: arithmetic mean
zbMATH Keywords: geometric mean
zbMATH Keywords: harmonic mean
zbMATH Keywords: differential subordination
zbMATH Keywords: convex function
MaRDI profile type: Publication
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OpenAlex ID: W2895960269
Wikidata QID: Q129063833
description: scientific article; zbMATH DE number 6958788