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Proceedings of the American Mathematical Society

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

 

 

Level sets for functions convex in one direction


Author: Johnny E. Brown
Journal: Proc. Amer. Math. Soc. 100 (1987), 442-446
MSC: Primary 30C45
DOI: https://doi.org/10.1090/S0002-9939-1987-0891142-9
MathSciNet review: 891142
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Abstract: Goodman and Saff conjectured that if $ f$ is convex in the direction of the imaginary axis then so are the functions $ \frac{1}{r}f(rz)$ for all $ 0 < r < \sqrt 2 - 1$, i.e., the level sets $ f(\vert z\vert\, < r)$ are convex in the direction of the imaginary axis for $ 0 < r < \sqrt 2 - 1 $. A weak form of this conjecture is proved and a question of Brannan is answered negatively.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0891142-9
Article copyright: © Copyright 1987 American Mathematical Society