Properties for Generalized Differential of (h, φ)-convex Functions and (h, φ)-Lipschitz Function

By making use of the relationship between a function and f its corresponding function f(t)= φ(f(h^-1(t)),this paper studied some properties for generalized directional derivatives of (h,φ)-convex functions and (h,φ)-Lipschitz functions.It is shown that generalized directional derivative of a continuous (h,φ)-convex function defined on Rⁿ is finite,upper semicontinuous and satisfies an inequality.A necessary and sufficient condition characterizing (h,φ)-Lipschitz functions f defined on Rⁿ is obtained under the assumption that f is (h,φ)-convex.As applications,the relation between (h,φ)-convex functions and (h,φ)-Lipschitz functions,and some fundamental properties of the generalized subdifferential of (h,φ)-convex functions are presented.