The uniqueness of meromorphic functions with their derivatives

In this paper, we deal with the problem of uniqueness of meromorphic functions that share one finite value with their derivatives and obtain some theorems which improve a result given by Rainer Bruck. Introduction and main resultsBy a meromorphic function we shall always mean a function that is meromorphic in the whole complex plane.It is assumed that the reader is familiar with the notations of the Nevanlinna theory such as Γ(r,/), ra(r,/), JV(r,/), N(r,f), S(r,f) and so on, that can be found, for instance, in [1].And Ni)(r, I//) denotes the counting function of the simple zeros of/, Npfr, I//) = N(r, I//) -Λ/Ί)(r, I//).Let / and g be meromorphic functions and a be a complex constant, we say that / and g share the value a IM (ignoring multiplicity), if /a and g -a have the same zeros, they share the value a CM (counting multiplicity), if /a and g -a have the same zeros with the same multiplicity.In 1979, E. Mues and N. Steinmetz proved the following theorem in [2].THEOREM A. Let f be an entire function which is not constant.Iff and f share two distinct values a, b, then f = f.In 1996, Rainer Briick proved the following in [3] THEOREM B. Let f be an entire function which is not constant.If f and f share the value 1 CM, and if N(r, I// 7 ) = S(r,/), then £Ξί « for some non-zero constant c.