On the derivative of meromorphic functions with multiple zeros ✩

Let f be a transcendental meromorphic function and let R be a rational function, R ≡ 0. We show that if all zeros and poles of f are multiple, except possibly finitely many, then f − R has infinitely many zeros. If f has finite order and R is a polynomial, then the conclusion holds without the hypothesis that poles be multiple.  2003 Elsevier Science (USA). All rights reserved.