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Proceedings of the American Mathematical Society

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



The starlike radius for classes of regular bounded functions

Author: Robert W. Sanders
Journal: Proc. Amer. Math. Soc. 54 (1976), 217-220
MathSciNet review: 0390201
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Abstract: Let $ {B_0}(a)$ be the class of all functions $ f$ defined on $ \vert z\vert < 1$ such that (i) $ f(z)$ is regular, (ii) $ \vert f(z)\vert < 1$ (iii) $ f(0) = 0$ (iv) $ 0 < \vert f'(0)\vert = a \leqslant 1$. For fixed $ R,a \leqslant R < 1$, let $ {B_0}(a;R)$ be that subclass having nonzero zeros at $ z = {z_k},k = 1,2, \ldots $, such that $ \prod \vert{z_k}\vert = R$. The subclass having no nonzero zeros is designated as $ {B_0}(a;1)$. A sharp lower bound for $ \operatorname{Re} [zf'(z)/f(z)]$ for the class $ {B_0}(a;R),a \leqslant R \leqslant 1$, is obtained, and the radius of starlikeness is found. A covering theorem for the class is also obtained.

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Keywords: Starlike radius, bounded functions
Article copyright: © Copyright 1976 American Mathematical Society