Two meromorphic functions f and g share the value a by counting multiplicities (CM) if the zeros of f-a and g-a (1/f and 1/g if are the same with same multiplicities.
In the paper under review the following is proved: Theorem 1. Let g be an entire function which shares some finite nonzero value a CM with its derivative g’. If, moreover, whenever , then
Theorem 2. Let f be meromorphic in the plane and assume that f, f’ and f” share the finite value CM. Then . The example shows that the assumption may not be dropped. It is, however, open whether Theorem 1 holds also for meromorphic functions.
The proof of Theorem 2 is quite extensive. It requires to construct highly artistic auxiliary functions, such as
and leads finally to the surprising contradiction .