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Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

Walter Bergweiler Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Alexandre Eremenko Department of Mathematics, Purdue University

TBD mathscidoc:1701.331977

Acta Mathematica, 197, (2), 145-166, 2005.12
We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the$n$th derivative tends to infinity, as $$n\to\infty$$ . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.
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@inproceedings{walter2005proof,
  title={Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions},
  author={Walter Bergweiler, and Alexandre Eremenko},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203349823184686},
  booktitle={Acta Mathematica},
  volume={197},
  number={2},
  pages={145-166},
  year={2005},
}
Walter Bergweiler, and Alexandre Eremenko. Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions. 2005. Vol. 197. In Acta Mathematica. pp.145-166. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203349823184686.
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