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 | References | Similar Articles | Additional Information

Abstract: Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then 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

DOI:
https://doi.org/10.1090/S0002-9939-96-03083-3

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