This is an announcement for the paper "The weak Banach-Saks Property of the Space $(L_\mu^p)^m$" by Zhenglu Jiang and Xiaoyong Fu. Abstract: 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. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20, 40H05, 40G05, 47F05 Remarks: 7 The source file(s), jf-bs.tex: 29847 bytes, is(are) stored in gzipped form as 0702537.gz with size 8kb. The corresponding postcript file has gzipped size 104kb. Submitted from: mcsjzl@mail.sysu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0702537 or http://arXiv.org/abs/math.FA/0702537 or by email in unzipped form by transmitting an empty message with subject line uget 0702537 or in gzipped form by using subject line get 0702537 to: math@arXiv.org.