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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the radius of starlikeness of $(zf)^{’}$ for $f$ univalent
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by Roger W. Barnard PDF
Proc. Amer. Math. Soc. 53 (1975), 385-390 Request permission

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 $|z| < 1$, 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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 53 (1975), 385-390
  • MSC: Primary 30A32
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0382615-8
  • MathSciNet review: 0382615