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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarks on a multiplier conjecture for univalent functions


Authors: Richard Fournier and Stephan Ruscheweyh
Journal: Proc. Amer. Math. Soc. 116 (1992), 35-43
MSC: Primary 30C45
MathSciNet review: 1101983
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Abstract: In this paper we study some aspects of a conjecture on the convolution of univalent functions in the unit disk $ \mathbb{D}$, which was recently proposed by Grünberg, Rønning, and Ruscheweyh (Trans. Amer. Math. Soc. 322 (1990), 377-393) and is as follows: let $ \mathcal{D}: = \{ f{\text{ analytic in }}\mathbb{D}:\left\vert {f''(z)} \right\vert \leq \operatorname{Re} f'(z),z \in \mathbb{D}\} $ and $ g,h \in \mathcal{S}$ (the class of normalized univalent functions in $ \mathbb{D}$. Then $ \operatorname{Re} (f*g*h)(z)/z > 0$ in $ \mathbb{D}$. We discuss several special cases, which lead to interesting, more specific statements about functions in $ \mathcal{S}$, determine certain extreme points of $ \mathcal{D}$, and note that the former conjectures of Bieberbach and Sheil-Small are contained in this one. It is an interesting matter of fact that the functions in $ \mathcal{D}$, which are "responsible" for the Bieberbach coefficient estimates are not extreme points in $ \mathcal{D}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1101983-2
PII: S 0002-9939(1992)1101983-2
Article copyright: © Copyright 1992 American Mathematical Society