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A counterexample to the two-thirds conjecture


Authors: Roger W. Barnard and John L. Lewis
Journal: Proc. Amer. Math. Soc. 41 (1973), 525-529
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1973-0325944-4
MathSciNet review: 0325944
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Abstract: Let $ w = f(z) = z + {a_2}{z^2} + \cdots $ be regular and univalent for $ \vert z\vert < 1$, and $ \operatorname{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 $ f(z),{d^\ast } = \min \vert f(z)\vert$ for $ \vert z\vert = {r_0}$ and $ d = \inf \vert\beta \vert$ for which $ f(z) \ne \beta $, then it has been conjectured by A. Schild in 1953 that $ {d^\ast }/d \geqq \tfrac{2}{3}$. It is shown here that this conjecture is false by giving two counter-examples.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0325944-4
Keywords: Univalent functions, starlike functions, convex functions, radius of convexity
Article copyright: © Copyright 1973 American Mathematical Society

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