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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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A note on a problem of Robinson
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by Kent Pearce PDF
Proc. Amer. Math. Soc. 89 (1983), 623-627 Request permission

Abstract:

Let $\mathcal {S}$ be the usual class of univalent analytic functions on $\left | z \right | < 1$ normalized by $f(0) = 0$ and $f’(0) = 1$. Let $\mathfrak {L}$ be the linear operator on $\mathcal {S}$ given by $\mathfrak {L}f = \tfrac {1}{2}(zf)’$ and let ${r_{{\mathcal {S}_t}}}$ be the minimum radius of starlikeness of $\mathfrak {L}f$ for $f$ in $\mathcal {S}$. In 1947 R. M. Robinson initiated the study of properties of $\mathfrak {L}$ acting on $\mathcal {S}$ when he showed that ${r_{{\mathcal {S}_t}}} > .38$. Later, in 1975, R. W. Barnard gave an example which showed ${r_{{\mathcal {S}_t}}} < .445$. It is shown here, using a distortion theorem and Jenkin’s region of variability for $zf’(z)/f(z)$, $f$ in $\mathcal {S}$, that ${r_{{\mathcal {S}_t}}} > .435$. Also, a simple example, a close-to-convex half-line mapping, is given which again shows ${r_{{\mathcal {S}_t}}} < .445$.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 89 (1983), 623-627
  • MSC: Primary 30C45
  • DOI: https://doi.org/10.1090/S0002-9939-1983-0718985-2
  • MathSciNet review: 718985