Generalized Jordan derivations of unital algebras

Let A be a unital algebra over a field F with char(F) not equal 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 y; h : A -> A such that f (x) o y + x o y (y) = h (x o y) for all x, y E A (here x o 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.