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Proceedings of the American Mathematical Society

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

 

 

On the radius of starlikeness of $ (zf)\sp{'} $ for $ f$ univalent


Author: Roger W. Barnard
Journal: Proc. Amer. Math. Soc. 53 (1975), 385-390
MSC: Primary 30A32
MathSciNet review: 0382615
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Abstract: Let $ S$ be the standard class of normalized univalent functions. For a given function $ f,\;f(z) = z + {a_2}{z^2} + \ldots $, regular for $ \vert z\vert < 1$, let $ r(f)$ be the radius of starlikeness of $ f$. In 1947, R. M. Robinson considered the combination $ {g_f}(z) = (zf)'/2$ for $ f\epsilon S$. He found a lower bound of .38 for $ r({g_f})$ for all $ f\epsilon S$. He noted that the standard Koebe function $ k,\;k(z) = z{(1 - z)^2}$, has its $ r({g_k})$ equal to $ 1/2$. A question that has been asked since Robinson's paper is whether $ 1/2$ is the minimum $ r({g_f})$ for all $ f$ in $ S$. It is shown here that this is not the case by giving examples of functions $ f$ whose $ r({g_{f}})$ is less than $ 1/2$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0382615-8
Keywords: Univalent functions, starlike function, radius of starlikeness
Article copyright: © Copyright 1975 American Mathematical Society