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

Abstract

Let f be a real meromorphic function of infinite order in the plane, with finitely many zeros and non-real poles. Then f″ has infinitely many non-real zeros.

The reciprocal of a real entire function and non-real zeros of higher derivatives

“…The case m = 0 is already contained in Theorem 1.1, but the present proof is somewhat simpler than that in [32]. The final result is linked to the investigations of [28,30,31,32,33], which in turn followed on from earlier work [18,19,20,21,22,37] concerning the existence of non-real zeros of derivatives of real meromorphic functions in general. It seems likely that if k ≥ 2 and f is a real meromorphic function in the plane, such that f and f (k) have finitely many non-real zeros, then f has in some sense relatively few distinct non-real poles.…”

“…The starting point of the present paper is the analogous problem where f , rather than being entire, is the reciprocal of a real entire function with real zeros. The following theorem [31,32] represents a strengthening of results of Hellerstein and Williamson [18] and Rossi [37]. Here N N R (r, g) denotes the Nevanlinna counting function [12] of the non-real poles of a meromorphic function g in the plane, and this notation will be used throughout the paper.…”

On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities

“…in the case when the entire function f (z) has only finitely many non-real zeroes. The zeroes of the function F κ [f ](z) when f (z) is a meromorphic function were studied in [2,17,18,19,27].…”

Non-real zeros of derivatives of meromorphic functions

“…Here f is called real if f = f , and strictly non-real if f is not a constant multiple of f . There has been substantial research concerning non-real zeros of derivatives of real entire or real meromorphic functions [2,3,5,14,15,16,20,24,25,26,30,32], but somewhat less in the strictly non-real case. The following theorem was proved in [13].…”