Let A be a *-algebra, D :A-->A be a linear map, and z in A be fixed. We consider the condition that D satisfies xD(y)*+D(x)y*=D(z) ( x*D(y)+D(x)*y=D(z) whenever xy*=z (x*y=z), and under several conditions on A, D and z we characterize the structure of D. In particular, we prove that if A is a Banach *-algebra, D is a continuous linear map, and z is a left (right) separating point of A, then D is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map D satisfying the above conditions with z=0 on two classes of *-algebras: zero product determined algebras and standard operator algebras.
