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Proceedings of the American Mathematical Society

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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}$.


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

  • [1] P. Erdös, F. Herzog and G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125-148. MR 21 #123. MR 0101311 (21:123)
  • [2] Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), 97-115. MR 27 #1564. MR 0151580 (27:1564)
  • [3] Dorothy Browne Schaffer, Distortion theorems for lemniscates and level loci of Green's functions, J. Analyse Math. 17 (1966), 59-70. MR 36 #361. MR 0217270 (36:361)

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Additional Information

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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