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Critical points of real entire functions and a conjecture of Pólya

Authors:	Young-One Kim

Institution:	Department of Mathematics, College of Natural Sciences, Sejong University, Seoul 133--747, Korea

Abstract:	Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip $\|\operatorname{Im} z\|\leq A$ , . It is shown that (1) if has nonreal zeros in the region $a\leq \operatorname{Re} z \leq b$ , 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 $\rho$ , $0<\rho \leq 2$ , and minimal type, then for each positive constant there is a positive integer 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 is of order less than , then has just as many critical points as couples of nonreal zeros.

Keywords:	P\'{o}lya--Wiman conjecture Laguerre--P\'{o}lya class Fourier critical point

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