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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on a problem of Robinson


Author: Kent Pearce
Journal: Proc. Amer. Math. Soc. 89 (1983), 623-627
MSC: Primary 30C45
MathSciNet review: 718985
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Abstract: Let $ \mathcal{S}$ be the usual class of univalent analytic functions on $ \left\vert z \right\vert < 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

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0718985-2
PII: S 0002-9939(1983)0718985-2
Keywords: Univalence, radius of starlikeness, Robinson's $ \tfrac{1}{2}$ conjecture
Article copyright: © Copyright 1983 American Mathematical Society