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Proceedings of the American Mathematical Society

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



A note on the $ 2/3$ conjecture for starlike functions

Authors: Carl P. McCarty and David E. Tepper
Journal: Proc. Amer. Math. Soc. 34 (1972), 417-421
MSC: Primary 30A32
MathSciNet review: 0304632
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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 0.380 \cdots $ which improves the old estimate $ {d_0}/{d^\ast} \geqq 0.343 \cdots $. In addition an upper bound for $ {d^ \ast }$ which depends on $ \vert{a_2}\vert$ is given.

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Keywords: Schlicht functions, convex functions, starlike functions, radius of convexity
Article copyright: © Copyright 1972 American Mathematical Society