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Critical points of real entire functions
and a conjecture of Pólya

Author: Young-One Kim
Journal: Proc. Amer. Math. Soc. 124 (1996), 819-830
MSC (1991): Primary 30D15, 30D35
MathSciNet review: 1301508
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Abstract: Let $f(z)$ be a nonconstant real entire function of genus $1^*$ and assume that all the zeros of $f(z)$ are distributed in some infinite strip $|\operatorname{Im} z|\leq A$, $A>0$. It is shown that (1) if $f(z)$ has $2J$ nonreal zeros in the region $a\leq \operatorname{Re} z \leq b$, and $f'(z)$ has $2J'$ nonreal zeros in the same region, and if the points $z=a$ and $z=b$ are located outside the Jensen disks of $f(z)$, then $f'(z)$ has exactly $J-J'$ critical zeros in the closed interval $[a,b]$, (2) if $f(z)$ is at most of order $\rho $, $0<\rho \leq 2$, and minimal type, then for each positive constant $B$ there is a positive integer $n_1$ such that for all $n\geq n_1$ $f^{(n)}(z)$ has only real zeros in the region $|\operatorname{Re} z|\leq Bn^{1/\rho }$, and (3) if $f(z)$ is of order less than $2/3$, then $f(z)$ has just as many critical points as couples of nonreal zeros.

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Additional Information

Young-One Kim
Affiliation: Department of Mathematics, College of Natural Sciences, Sejong University, Seoul 133–747, Korea

Keywords: P\'{o}lya--Wiman conjecture, Laguerre--P\'{o}lya class, Fourier critical point
Received by editor(s): March 28, 1994
Received by editor(s) in revised form: September 7, 1994
Additional Notes: This research is supported by the research grant of the Ministry of Education, Republic of Korea, and SNU–GARC.
Dedicated: To the memory of Professor Jongsik Kim
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1996 American Mathematical Society

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