Generalized Jordan derivations of unital algebras

Let A be a unital algebra over a field F with char(F) ? 2. In this paper we introduce a new concept of a generalized Jordan derivation, covering Jordan centralizers and Jordan derivations, as follows: a linear map f : A ? A is a generalized Jordan derivation if there exist linear maps 1; h : A ? A such that f (x) ? y + x ? g (y)= h (x ? y) for all x, y ? A (here x ? y = xy + yx). Our aim is to give the form of map f in terms of the so called quasi Jordan centralizers and quasi Jordan derivations. In addition, a characterization of such maps is presented.