Generalized Zalcman conjecture for starlike and typically real functions
type: Item
instance of: scholarly article
title: Generalized Zalcman conjecture for starlike and typically real functions
zbMATH Open document ID: 0936.30012
author: William Ma
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-05-15T00:00:00Z
review text: Let \(S\) be the class of normalized univalent functions, i.e. those functions of the form \(f(z) = z + \sum_{n=1}^\infty a_n z^n\) which are analytic and univalent in the unit disc. Zalcman conjectured that if \(f \in S\) then \(|a_n^2 - a_{2n-1}|\leq (n -1)^2\) for all \(n \geq 2\). This is clearly difficult to prove since it implies the Bieberbach conjecture. (The proof of the latter does not seem to offer any help in the proof of the Zalcman conjecture.) The Zalcman conjecture has been proved true for starlike functions, for functions in \(S\) with real coefficients, for typically real functions, and for close-to-convex functions when \(n \geq 4\). In this paper, the author offers the generalized Zalcman conjecture: \(f \in S\) implies \(|a_n a_m - a_{n+m-1}|\leq (n-1)(m-1)\) for all \(m,n \geq 2\). This is proved true for starlike functions and for functions with real coefficients. The latter result is proved with the help of the result, also proved in the paper, that the generalized Zalcman conjecture holds for typically real functions when \(n,m \geq 4\).
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1309062
reviewed by: James A. Hummel
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6378
OpenAlex ID: W2058036837
Wikidata QID: Q122942784
cites work: Convex Hulls of Some Classical Families of Univalent Functions
cites work: On the Zalcman conjecture for starlike and typically real functions
cites work: The Coefficients of Multivalent Close-to-Convex Functions
cites work: The Zalcman Conjecture for Close-to-Convex Functions
cites work: Q4052427
DOI: 10.1006/JMAA.1999.6378
description: scientific article; zbMATH DE number 1309062