## Remarks on the Obrechkoff inequality

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- by Alexandre Eremenko and Alexander Fryntov
- Proc. Amer. Math. Soc.
**144**(2016), 703-707 - DOI: https://doi.org/10.1090/proc/12738
- Published electronically: August 20, 2015
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## Abstract:

Let $u$ be the logarithmic potential of a probability measure $\mu$ in the plane that satisfies \[ u(z)=u(\overline {z}),\quad u(z) \le u(|z|),\quad z\in \mathbb {C},\] and $m(t)=\mu \{ z\in \mathbb {C}^*:|\operatorname {Arg} z|\leq t\}$. Then \[ \frac {1}{a}\int _0^a m(t)dt\leq \frac {a}{2\pi },\] for every $a\in (0,\pi )$. This improves and generalizes a result of Obrechkoff on zeros of polynomials with positive coefficients.## References

- 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**, DOI 10.1090/S0002-9939-1991-1072329-2 - 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**, DOI 10.5186/aasfm.2015.4022 - S. Ghosh and O. Zeitouni,
*Large deviations for zeros of random polynomials with i.i.d. exponential coefficients*, arXiv:1312.6195. - 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** - N. Obrechkoff,
*Sur un problème de Laguerre*, C. R. Acad. Sci. (Paris)**177**(1923), 102–104. - H. Poincaré,
*Sur les équations algébriques*, C. R. Acad. Sci.**97**(1884), 1418–1419. - O. Zeitouni,
*Zeros of polynomials with positive coefficients*, http://mathoverflow.net/questions/134998.

## Bibliographic Information

**Alexandre Eremenko**- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 63860
**Alexander Fryntov**- Affiliation: Physical-Engineering Institute of Low Temperature, National Academy of Sciences of Ukraine, Kharkov 310164, Ukraine
- MR Author ID: 190280
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 703-707 - MSC (2010): Primary 30C15, 31A05
- DOI: https://doi.org/10.1090/proc/12738
- MathSciNet review: 3430846