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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Generating functions for some classes of univalent functions

Authors: Zdzisław Lewandowski, Sanford Miller and Eligiusz Złotkiewicz
Journal: Proc. Amer. Math. Soc. 56 (1976), 111-117
MathSciNet review: 0399438
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Abstract: Let $ P(z) = {e^{i\beta }} + {p_1}z + {p_2}{z^2} + \cdots $ be regular in the unit disc $ \Delta $ with $ \vert\beta \vert < \pi /2$, and let $ \psi (u,v)$ be a continuous function defined in a domain of $ {\mathbf{C}} \times {\mathbf{C}}$. With some very simple restrictions on $ \psi (u,v)$ the authors prove a lemma that $ \operatorname{Re} \psi (p(z),zp'(z)) > 0$ implies $ \operatorname{Re} p(z) > 0$. This result is then used to generate subclasses of starlike, spirallike and close-to-convex functions.

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Keywords: Functions with positive real part, Carathéodory functions, univalent functions, starlike functions, spirallike functions, close-to-convex functions
Article copyright: © Copyright 1976 American Mathematical Society