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Proceedings of the American Mathematical Society
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0390201-X
PII: S 0002-9939(1976)0390201-X
Keywords: Starlike radius, bounded functions
Article copyright: © Copyright 1976 American Mathematical Society