In this paper, we introduce Bα, β-operator as a new techniqne to generalize Fitzpatrick functions. We study the properties of this new operator, such as convexity, weak∗-closedness, locally boundedness and Fenchel duality. Furthermore, we present new concept that is Cα n-monotone bifunction. By the application of the new operator, we prove that ψβ, n (x, x∗)=〈 x∗, x〉∀(x, x∗)∈ G (Bα, β), for each n≥ 2. This equation is a generalization for a known result which is studied by many researchers.
