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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the radius of convexity and boundary distortion of Schlicht functions


Author: David E. Tepper
Journal: Trans. Amer. Math. Soc. 150 (1970), 519-528
MSC: Primary 30.42
MathSciNet review: 0268370
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Abstract: Let $ w = f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}} $ be regular and univalent for $ \vert z\vert < 1$ and map $ \vert z\vert < 1$ onto a region which is starlike with respect to $ w = 0$. If $ {r_0}$ denotes the radius of convexity of $ w = f(z)$, $ d_0 = \min \vert f(z)\vert$ for $ \vert z\vert = {r_0}$, and $ {d^ \ast } = \inf \vert\beta \vert$ for $ f(z) \ne \beta $, then it has been conjectured that $ {d_0}/{d^ \ast } \geqq 2/3$. It is shown here that $ {d_0}/{d^ \ast } \geqq 0.343 \ldots $, which improves the old estimate $ {d_0}/{d^ \ast } \geqq 0.268 \ldots $. In addition, sharp estimates for $ {r_0}$ are given which depend on the value of $ \vert{a_2}\vert$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0268370-0
PII: S 0002-9947(1970)0268370-0
Keywords: Schlicht functions, convex and starlike functions, radius of convexity
Article copyright: © Copyright 1970 American Mathematical Society