A simple proof of inequalities of integrals of composite functions

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space Rm and vector-valued functions in a weakly compact subset of a Banach vector space generated by m -spaces for 1⩽p<+∞. Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m -spaces instead.