  In this paper we show the weak Banach-Saks property of the Banach vector
space $(L_\mu^p)^m$ generated by $m$ $L_\mu^p$-spaces for $1\leq p<+\infty,$
where $m$ is any given natural number. When $m=1,$ this is the famous
Banach-Saks-Szlenk theorem. By use of this property, we also present
inequalities for integrals of functions that are the composition of nonnegative
continuous convex functions on a convex set of a vector space ${\bf R}^m$ and
vector-valued functions in a weakly compact subset of the space $(L_\mu^p)^m$
for $1\leq p<+\infty$ and inequalities when these vector-valued functions are
in a weakly* compact subset of the product space $(L_\mu^\infty)^m$ generated
by $m$ $L_\mu^\infty$-spaces.
