On the dependence of the growth of an entire function on the distribution of the zeros of its derivatives (On a question of G. Pólya and A. Wiman)

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

“…The transcendental function L = f ′ /f again has a Levin-Ostrovskii factorisation (14), in which P and ψ are real meromorphic functions and P has finitely many poles [34,35]. In this setting, if f has finitely many poles then ψ = 1, while if f has infinitely many poles then ψ is an infinite product constructed exactly as in Section 5.…”

“…Extensive research into the non-real zeros of derivatives of real entire functions [2,3,4,5,7,16,17,23,24,25,35,38] arose largely from the conjecture of Wiman (now proved in [3,35,38]) that if f is a real entire function such that f and f ′′ have only real zeros, then f belongs to the Laguerre-Pólya class consisting of locally uniform limits of real polynomials with real zeros, as well as from related conjectures of Pólya [36]. 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.…”

Equi-Distribution of Values for the Third and the Fifth Painlevé Transcendents

“…We review several facts on value distribution theory in a half plane ( [6], [21], [22] For each θ ∈ [0, 2π], applying Lemma 4.1 to f (z) − e iθ , we have…”

Reality of the zeros of an entire function and its derivatives

“…The infinite order case has remained open until now. Partial results have been obtained by Edrei [3], Levin and Ostrovskii [10], Hellerstein [5], and Hellerstein and Yang [9].…”

“…where {rke"Pt) are the poles of f(z) and n0(t, 00) denotes the number of poles of/in the region (| z -it/2\< t/2, \z\> 1}. Then the corresponding characteristic function for/is defined by The initial part of our proof is based on the work of Levin and Ostrovskii [10]. The Tsuji functionals are central to their proofs.…”