Entire functions and their first derivatives sharing simple β-points for a small function β

Abstract

The following theorem has been proved by A. Schweizer [7]. If a nonconstant entire function f and its derivative f′ share their simple zeros and if every simple a-point of f is a (not necessarily simple) a-point of f′ for some nonzero constant a, then f ≡ f′. In this paper we shall prove that the above result is also true when the nonzero constant a is replaced by a meromorphic small function β( $\not\equiv$ 0, ∞).