Real zeros of derivatives of meromorphic functions and solutions of second order differential equations

Abstract

We classify all functions F meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that F' has no zeros and F" only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation w" + H(z)w = 0, H entire. Introductionand statement of the main results. In a series of papers [3, 4, 6] the authors recently settled an old conjecture of Pólya by characterizing those entire functions / for which /, /' and /" have only real zeros. These results may be summarized by

Non-real zeros of derivatives of meromorphic functions

“…It follows from (1.4) that a zero of β is a pole of f and hence of f ′′ /f , while a pole of β is a zero of f or f : thus if f has only real zeros and f ′′ /f is entire then β has neither zeros nor poles, and so Theorem 1.3 contains [19,Theorem 5]. Observe further that if β is a real entire function with real zeros, all of even multiplicity, then (1.4) defines a strictly non-real meromorphic function f with real poles and no zeros, such that f ′′ /f is real.…”

“…It is known [3,47] that if f is a real transcendental entire function then f and f ′′ have only real zeros if and only if f belongs to the Laguerre-Pólya class LP , consisting of all entire functions which are locally uniform limits of real polynomials with real zeros, in which case all derivatives of f have only real zeros. For the real meromorphic case, the following was conjectured in [19]. Conjecture 1.1 ( [19]) Let f be a real transcendental meromorphic function in the plane with at least one pole, and assume that all zeros and poles of f , f ′ and f ′′ are real, and that all poles of f are simple.…”

“…For the real meromorphic case, the following was conjectured in [19]. Conjecture 1.1 ( [19]) Let f be a real transcendental meromorphic function in the plane with at least one pole, and assume that all zeros and poles of f , f ′ and f ′′ are real, and that all poles of f are simple. Then f satisfies f (z) = C tan(az + b) + Dz + E, a, b, C, D, E ∈ R.…”

“…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).…”

“…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].…”