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 continous convex functions on a vector space ${\mathbf{R}}^m$ and vector-valued functions in a weakly compact subset of a Banach vector space generated by $m$ $L_{\mu }^p$-spaces for $1\leq p<+\infty.$ Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by $m$ $L_{\mu }^{\mathrm{\infty }}$-spaces instead.