In the present note, two Opial-type integral inequalities involving many functions in several independent variables are proved.
Let be a rectangular region. Denote by a general point in and write . The main result embodied in Theorem 1 can be re-stated as follows.
Theorem 1. Let and their partial derivatives , and are all defined and continuous on . Let further be any nonnegative differentiable functions, , with nonnegative, continuous and nondecreasing on . Suppose that
for all , . Then we have
Another inequality given in Theorem 2 is obtained under the additional condition
Some known inequalities due to G. S. Yang [Tamkang J. Math. 13, 255-259 (1982; Zbl 0516.26009)] established for two-variable functions are contained in the results obtained.