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

被引：19

作者：

Bergweiler, Walter ^{[1
]}

Eremenko, Alexandre

机构：

[1] Univ Kiel, Math Seminar, DE-24098 Kiel, Germany

[2] Purdue Univ, Dept Math, W Lafayette, IN 47906 USA

来源：

ACTA MATHEMATICA | 2006年 / 197卷 / 02期

基金：

美国国家科学基金会;

关键词：

D O I：

10.1007/s11511-006-0010-8

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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 ntoinfty. 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. © 2007 Institut Mittag-Leffler.

引用

页码：145 / 166

页数：22

共 32 条

[11] NON-REAL ZEROS OF REAL ENTIRE FUNCTIONS
GALLAGHER, PX
[J]. MATHEMATISCHE ZEITSCHRIFT, 1965, 90 (02) : 124 - +
[12] ON THE ZEROS OF THE DERIVATIVES OF SOME ENTIRE-FUNCTIONS OF FINITE-ORDER
GETHNER, RM
[J]. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 1985, 28 : 381 - 407
[13] HAYMAN WK, 1956, J REINE ANGEW MATH, V196, P67
[14] DERIVATIVES OF ENTIRE FUNCTIONS AND A QUESTION OF POLYA .2.
HELLERSTEIN, S
WILLIAMSON, J
[J]. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1977, 234 (02) : 497 - 503
[15] DERIVATIVES OF ENTIRE FUNCTIONS AND A QUESTION OF POLYA
HELLERSTEIN, S
WILLIAMSON, J
[J]. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1977, 227 (MAR) : 227 - 249
[16] HELLERSTEIN S, 1971, J LOND MATH SOC, V4, P469
[17] Hirschman II., 1955, The Convolution Transform
[18] Hormander L., 1990, Distribution theory and Fourier analysis, VI
[19] Kac I. S., 1974, Amer. Math. Soc. Transl., V103, P1
[20] Karlin S., 1968, TOTAL POSITIVITY, V1

← 1 2 3 4 →