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 R m and vector-valued functions in a weakly compact subset of a Banach vector space generated by m L μ p -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 L μ ∞ -spaces instead.