OPIAL-TYPE INEQUALITIES WITH m FUNCTIONS IN n VARIABLES WING-SUM CHEUNG Abstract. In this paper, the Opial's inequality, which has a wide range of applications in the study of differential and integral equations, is generalized to the case involving m functions of n variables, m, nsl, §1. Introduction. The integrodifferential inequalities in general play an important role in the study of qualitative as well as quantitative properties of solutions of many differential and integral equations. Among these the follow- ing Opial's inequality has continuously drawn people's attentions and has proven to be useful in many situations. THEOREM (Opial [7]). Iffe g l [0, h] satisfies/(0) =/(/») = 0 andf(x) > 0 forallxe(0,h), then h h J JJ \f'(x)\ 2 dx. Opial's inequality has been generalized to many different situations. For instance, in [2] and [10], it was generalized to the case involving two functions of one variable; and in [3] and [8], to the case involving two functions of two variables. In this paper, it is further generalized to the case involving m functions of n variables, where m, n & 1. §2. Main results. Throughout the paper we let n> 1 and m ? 2 be any two fixed integers. Let a,{},... be indices running from 1 to m, and ij,... from 1 to n. For each a, let / " be a continuous real-valued function on a rectangular region ft = n"=i i a i, &,] C R" such that the partial derivatives fi,fi2, • • • ,f"...n are all defined and continuous on ft. For the sake of sim- plicity, we shall denote /"..„ by./". Let w be a positive continuous weight function on ft and r be any positive function on ft with r~ l e ^'(ft). Denote by x = (x,,..., x n ) a general point in ft, dx the volume form dx x ... dx n , and V(ft) the volume of the region ft. THEOREM 1. If f(a x ,x 2 x n )=fi(x i ,a 2 ,x 3 ,...,x n ) = • • • = /l...(«-I)(*l> • • • » x n-l* a n) = 0 [MATHEMATIKA, 39 (1992), 319-326]