Real meromorphic functions and linear differential polynomials

Abstract

We determine all real meromorphic functions f in the plane such that f′ has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f(k)+Σj=0k−1αjf(j), in which k ⩾ 2 and the coefficients aj are real numbers with a0 ≠ 0.