中文摘要：

证明了由m个Lμp空间产生的Banach向量空间（Lμp）m的弱Banach-Saks性质,其中m是自然数, 1 p 〈＋∞.当m= 1时,这就是著名的Banach-Saks-Szlenk定理.运用该性质,还给出了定义在向量空间Rm的一个凸集上的非负连续凸函数与取值在空间（Lpμ）m的一个弱紧子集中的向量值函数的复合函数的积分不等式.当这些向量值函数属于由m个Lμ∞空间产生的积空间（Lμ∞）m的一个弱＊紧子集时,类似的积分不等式还是成立的.

英文摘要：

We show the weak Banach-Saks property of the Banach vector space（Lpμ）m generated by m Lpμ-spaces for 1≤p〈＋∞,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 Rm and vector-valued functions in a weakly compact subset of the space（Lpμ）m for 1≤〈＋∞ and inequalities when these vector-valued functions are in a weakly^＊ compact subset of the product space （Lμ^p）^m generated by m Lμ^p -spaces.