In this paper, by using the theory of generalized convexity functions we introduce and prove monotonicity of sequences of the forms (The equation is abbreviated) where f belongs to the classes of AG-convex (concave), HA-convex (concave), or HGconvex (concave) functions defined on suitable intervals, {an} is a given sequence and φ is a given function that satisfy some preset conditions. As a consequence, we obtain some generalizations of Alzer type inequalities.