Non-real zeros of real differential polynomials

Abstract

The main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles and such that certain combinations of derivatives of f have few non-real zeros.

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

“…Because f (z) may be replaced by f (−z) it suffices to consider large x on the positive real axis R + . The assertion concerning f ′′ /f was already proved in [31,Section 3], and that for f (4) (x)/f (x) will now be established by dividing into cases.…”

“…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.…”

“…On the other hand, Lemma 5.3 below shows that most of these zeros lie relatively close to the real axis. The strategy for proving Theorem 1.2 is to demonstrate (see Proposition 5.1 below) that if f is as in the hypotheses then f (4) (x)/f (x) is positive or infinite for all real x with |x| sufficiently large, as was already proved for f ′′ (x)/f (x) in [31]. Indeed, it seems reasonable to conjecture that the same would be true for f (n) (x)/f (x), for any even positive integer n, but this appears to be difficult to prove.…”

“…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.…”

Non-real zeros of derivatives of meromorphic functions

“…Examples (III), (IV) and (V) arising from Theorem 1.1 show that (1.1) is not far from being sharp and that, at least for m = 2, the hypothesis that f has finite order is not redundant in the second assertion of Theorem 1.2. Note that the analogous problem when f is real was treated, but again not fully solved, in [20,35,37,45].…”