Meromorphic Functions That Share One Finite Value CM or IM with Their First Derivative

In this paper we shall prove that if a non-constant meromorphic function f and its derivative ƒ share the value a(= 0,æ) CM (IM) and if N^(r ,1/f)= S (r ,f) N^(r,1/F)+N^(R,1/f )=S(r,F)then either F=F or f (z)=A(z-c)/1+Ae^-z F(z)=2a/1-Ae^-2z where A(6= 0) and c are constants. These results give improvement and.extension of the following result of Gundersen: if a non-constant meromorphic function f and its derivative f share two distinct values 0, a(6=æ) CM, then f = f .