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Remarks on the Obrechkoff inequality


Authors: Alexandre Eremenko and Alexander Fryntov
Journal: Proc. Amer. Math. Soc. 144 (2016), 703-707
MSC (2010): Primary 30C15, 31A05
DOI: https://doi.org/10.1090/proc/12738
Published electronically: August 20, 2015
MathSciNet review: 3430846
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Abstract: Let $ u$ be the logarithmic potential of a probability measure $ \mu $ in the plane that satisfies

$\displaystyle u(z)=u(\overline {z}),\quad u(z) \le u(\vert z\vert),\quad z\in \mathbb{C},$

and $ m(t)=\mu \{ z\in \mathbb{C}^*:\vert\operatorname {Arg} z\vert\leq t\}$. Then

$\displaystyle \frac {1}{a}\int _0^a m(t)dt\leq \frac {a}{2\pi },$

for every $ a\in (0,\pi )$. This improves and generalizes a result of Obrechkoff on zeros of polynomials with positive coefficients.

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Additional Information

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Alexander Fryntov
Affiliation: Physical-Engineering Institute of Low Temperature, National Academy of Sciences of Ukraine, Kharkov 310164, Ukraine

DOI: https://doi.org/10.1090/proc/12738
Keywords: Polynomials, logarithmic potential
Received by editor(s): November 1, 2014
Received by editor(s) in revised form: January 22, 2015, and January 23, 2015
Published electronically: August 20, 2015
Additional Notes: This work was supported by NSF grant DMS-1361836.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2015 American Mathematical Society