  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 ${\bf R}^m$ 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 $m$
$L_\mu^\infty$-spaces instead.
