Acta Mathematica

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

Walter Bergweiler and Alexandre Eremenko

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Abstract

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 nth 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.

Article information

Source
Acta Math. Volume 197, Number 2 (2006), 145-166.

Dates
Received: 6 December 2005
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891863

Digital Object Identifier
doi:10.1007/s11511-006-0010-8

Zentralblatt MATH identifier
1121.30013

Rights
2007 © Institut Mittag-Leffler

Citation

Bergweiler, Walter; Eremenko, Alexandre. Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions. Acta Math. 197 (2006), no. 2, 145--166. doi:10.1007/s11511-006-0010-8. https://projecteuclid.org/euclid.acta/1485891863.


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