A proof of the Pólya-Wiman conjecture

Abstract

Abstract.Let f(z) = e~az g(z) where a > 0 and g is a real entire function of genus at most 1. It is shown that if / has only a finite number of nonreal zeros, then its derivatives, from a certain one onward, have only real zeros.

Non-real zeros of derivatives of real meromorphic functions of infinite order

“…This was proved by Sheil-Small [29] for f of finite order and in [6] for infinite order (see also [23]). Furthermore, for an entire function f = Ph, where h is a real entire function with real zeros and P is a real polynomial, the number of non-real zeros of f (k) is 0 for large k if h ∈ L P [7,8,16,17], and tends to infinity with k otherwise [5,18]: these results proved a conjecture of Pólya [26].…”

On the multiplicities of the zeros of Laguerre-Pólya functions

“…It may also be remarked that if ψ(z) is a transcendental Laguerre-Pólya function, then for each positive real number B there is a positive integer N such that ψ (n) (z) has only simple zeros in the interval [−B √ n, B √ n] whenever n ≥ N . For a proof of this fact, see [Km1]. Now, consider the functions ψ m (z), m = 1, 2, .…”

Critical points of real entire functions and a conjecture of Pólya

“…From Pólya's result mentioned above, we see that the hypothetical theorems B and C are equivalent. The hypothetical theorem C is known as the Pólya-Wiman conjecture and it has been completely proved by T. Craven, G. Csordas, W. Smith and the author [CCS1,CCS2,K1,K2], but it seems that no progress has been made on the hypothetical theorem A since 1930. Note that the hypothetical theorem A includes the assertion that the number of critical points and that of nonreal zeros are simultaneously infinite.…”