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

Abstract

Abstract. Two theorems are proved concerning non-real zeros of derivatives of the reciprocal of a real entire function with real zeros. A further result treats the frequency of non-real poles for real meromorphic functions which together with their first three derivatives have only real zeros.

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].…”

“…By [37,Lemma 2.2], there exist at most finitely many α ∈ C such that F (z) or L(z) tends to α as z tends to infinity along a path in H + . This makes it possible to choose θ ∈ (0, π) such that the two rays P ± , given respectively by w = te ±iθ , 0 < t < ∞, contain no critical values of L and no values α such that L(z) tends to α as z → ∞ along a path in H + .…”

“…Controlling the number of w-points of L in C, for w ∈ H + , requires a refinement of arguments from [36,37]. By [37,Lemma 2.2], there exist at most finitely many α ∈ C such that F (z) or L(z) tends to α as z tends to infinity along a path in H + .…”

“…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].…”

Non-real zeros of derivatives

“…is not new [22], but the present proof is considerably simpler than that of [22] and the result substantially more general. For results on non-real zeros of f ′′ + ωf when ω ≥ 0 and f has finite order, the reader is referred to [22,24,27] and [26,Theorem 1.5].…”