On the Zalcman conjecture for starlike and typically real functions
type: Item
instance of: scholarly article
title: On the Zalcman conjecture for starlike and typically real functions
zbMATH Open document ID: 0541.30004
DOI: 10.1007/BF01162720
author: Johnny E. Brown
author: Anna Tsao
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173688
review text: Let S denote the class of functions \(f(z)=z+a_ 2z^ 2+...\) which are analytic and univalent in the unit disc. Recently, L. Zalcman conjectured that \(| a^ 2_ n-a_{2n-1}| \leq(n-1)^ 2\) for all \(n\geq 2\) and each \(f\in S\) and showed that this inequality implies the Bieberbach conjecture. In this paper we investigate the functional \(\lambda a^ 2_ n-a_{2n-1},\) for certain real \(\lambda\), in the classes of typically real functions and starlike functions. This functional arises quite naturally when considering coefficient problems: If \(f\in S\) then the coefficients of \(f(z^ 2)^{{1\over2}}\) and 1/f(1/z) contain expressions of this form. For the class of typically real functions we give upper and lower bounds for \(\lambda a^ 2_ n-a_{2n-1},\) many of them sharp. The Zalcman conjecture is proved for typically real functions when \(n\geq 3\) and an immediate consequence is the truth of the conjecture for real functions in S when \(n\geq 2\). Our results also include a proof of the conjecture for starlike functions in S. We make use of integral representation formulas for these classes of functions together with an interesting coefficient inequality for functions of positive real part due to \textit{A. E. Livingston} [Proc. Am. Math. Soc. 21, 545-552 (1969; Zbl 0186.399)].
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 3859448
zbMATH Keywords: Bieberbach conjecture
zbMATH Keywords: typically real functions
zbMATH Keywords: starlike functions
zbMATH Keywords: Zalcman conjecture
Wikidata QID: Q123118562
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3859448