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Critical Points of Real Entire Functions and a Conjecture of Pólya

Young-One Kim
Proceedings of the American Mathematical Society
Vol. 124, No. 3 (Mar., 1996), pp. 819-830
Stable URL: http://www.jstor.org/stable/2161948
Page Count: 12
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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 ≤ ℜ z ≤ 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 n1 such that for all n ≥ n1 f(n) (z) has only real zeros in the region |ℜ z| ≤ Bn1/ρ, 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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