Non-real zeros of derivatives of real meromorphic functions

Abstract

Abstract. The main result of this paper determines all real meromorphic functions of finite order in the plane such that ′ has finitely many zeros while and ( ) , for some ≥ 2, have finitely many non-real zeros.

Non-real zeros of derivatives of meromorphic functions

“…While Conjecture 1.1 appears to be difficult to resolve in general, results proved in [19,27], and refined further in [33,34,43], show in particular that the conjecture is true subject to the additional hypothesis that f ′ omits some finite value, as is the case for the functions in (1.5). Theorems 1.4 and 1.5 below will resolve two further special cases of Conjecture 1.1, each of them linked to functions of the form (1.5).…”

“…Conversely, if f is given by (1.7) with R and the coefficients as in the last conclusion of Theorem 1.4, then f maps H + into itself, and all but finitely many zeros of f ′′ are real by [33,Lemma 3.2]. The next result in the direction of Conjecture 1.1 concerns the case where zeros of f ′′ are zeros of f ′ , as holds for example when f (z) = z − tan z. Theorem 1.5 Let f be a real transcendental meromorphic function in the plane such that: (a) all but finitely many zeros and poles of f and f ′ are real; (b) all but finitely many zeros of f ′′ are zeros of f ′ ; (c) the poles of f have bounded multiplicities; (d) either f has finitely many multiple poles, or f has finitely many simple poles.…”

“…Let m ≥ 3, denote positive constants by c j , and let w be a non-real zero of H (m) , the existence of which is assured by [33,Lemma 3.1]. Take a small positive t such 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 only other general result treating the strictly non-real case appears to be the following theorem from [18]. In the last example (VI) it is easy to verify that f is strictly non-real but f ′ is not, while f and g = f have no zeros, and the same poles, and f (m) and g (m) have the same zeros for all m ≥ 1; moreover, f ′ has no zeros, and f ′′ has only real zeros, but if m ≥ 3 then f (m) has infinitely many non-real zeros, by [33,Lemma 3.1]. The following theorem will be proved, and uses standard terminology from [14].…”