Real meromorphic functions and a result of Hinkkanen and Rossi

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 α and only finitely many nonreal zeroes, poles and α-points