Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

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

  • [1] R. W. Barnard, W. Dayawansa, K. Pearce, and D. Weinberg, Polynomials with nonnegative coefficients, Proc. Amer. Math. Soc. 113 (1991), no. 1, 77-85. MR 1072329 (91k:30007), https://doi.org/10.2307/2048441
  • [2] Walter Bergweiler and Alexandre Eremenko, Distribution of zeros of polynomials with positive coefficients, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 375-383. MR 3310090, https://doi.org/10.5186/aasfm.2015.4022
  • [3] S. Ghosh and O. Zeitouni, Large deviations for zeros of random polynomials with i.i.d. exponential coefficients, arXiv:1312.6195.
  • [4] B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888 (81k:30011)
  • [5] N. Obrechkoff, Sur un problème de Laguerre, C. R. Acad. Sci. (Paris) 177 (1923), 102-104.
  • [6] H. Poincaré, Sur les équations algébriques, C. R. Acad. Sci. 97 (1884), 1418-1419.
  • [7] O. Zeitouni, Zeros of polynomials with positive coefficients, http://mathoverflow.net/questions/134998.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30C15, 31A05

Retrieve articles in all journals with MSC (2010): 30C15, 31A05


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

American Mathematical Society