The Zeros of Derivatives of Entire Functions and the Pólya-Wiman Conjecture

A proof of the Pólya-Wiman conjecture

“…428]. This lemma is proved by using the same argument as in the original proof, except that a refinement of [CCS1,Lemma 3,p. 411] is now required.…”

“…On the other hand, in [Wil] and [Wi2], Wiman established the validity of this conjecture for functions of the form exp(-az +ßz)cp(z), where a > 0, ß € R, and cp is of genus zero. Recently, Craven, Csordas, and Smith proved the conjecture for functions of genus at most 1 [CCS1,CCS2]. Therefore the only remaining case of the conjecture is for functions of genus 2.…”

Non-real zeros of derivatives of meromorphic functions

“…Splitting the path Λ z into the part from i∞ to 4i|z| and the part Λ * z from 4i|z| to z now yields, for large z in S 1 , 5) and from (4.1) and (4.3) that…”

“…There has been extensive research into the existence of non-real zeros of derivatives of real entire or meromorphic functions [2,3,5,19,20,28,32,33,37,45,47], but rather less in the strictly non-real case. Meromorphic functions which, together with all their derivatives, have only real zeros were classified in [24,25,26].…”

The Lee‐Yang and Pólya‐Schur programs. II. Theory of stable polynomials and applications

“…This question has a distinguished history that goes back to Hermite, Laguerre, Hurwitz and Pólya-Schur, see [9] and references therein. In particular, in [41] Pólya and Schur characterized all diagonal operators with this property, which led to a rich subsequent literature on this subject [6,8,17,18,19,20,21,26,31,35,40,43,52]. However, it was not until very recently that full solutions to this question -and, more generally, to Problems 1-2 for n = 1 and any open circular domain Ω -were obtained in [6].…”