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Proceedings of the American Mathematical Society

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

 
 

 

On the convexity of lemniscates


Author: Dorothy Browne Shaffer
Journal: Proc. Amer. Math. Soc. 26 (1970), 619-620
MSC: Primary 30.11
DOI: https://doi.org/10.1090/S0002-9939-1970-0271313-2
MathSciNet review: 0271313
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Abstract: Let $ {L_1}$ denote the lemniscate $ \vert\prod\nolimits_{v = 1}^n {(z - {\zeta _v})\vert = 1} $. Assume the poles $ {\zeta _v}$ are inscribed in the disc $ \vert z\vert \leqq a$. Let $ {z_0} = {n^{ - 1}}\sum\nolimits_{v = 1{\zeta _v}}^n {} $. Conditions for the convexity of $ {L_1}$ are established in terms of $ a$ and $ {z_0}$. Sharp bounds are derived for real $ {\zeta _v}$.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0271313-2
Keywords: Lemniscate, level line, convexity
Article copyright: © Copyright 1970 American Mathematical Society

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