Real meromorphic functions and a result of Hinkkanen and Rossi

Abstract

Let $f$ be a transcendental meromorphic function such that all but finitely many of the poles of $f$ and zeroes of $f'$ are real. Generalising a result of Hinkkanen and Rossi (Proc. Amer. Math. Soc. 92 (1984) 72–74), we characterize those $f$ such that $f'$ takes some nonzero value only finitely often, and show that all but finitely many of the zeroes of $f''$ are real in this case. We also prove a related asymptotic result about real meromorphic functions with a nonzero deficient value $\alpha$ and only finitely many nonreal zeroes, poles and $\alpha$-points.