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 ${\bf R}^m$ 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. 
Zhenglu Jiang, Xiaoyong Fu and Hongjiong Tian