*(English)*Zbl 0761.26012

In the present note, two Opial-type integral inequalities involving many functions in several independent variables are proved.

Let ${\Omega}:=[{a}_{1},{b}_{1}]\times [{a}_{2},{b}_{2}]\times \cdots \times [{a}_{n},{b}_{n}]\subset {\mathbb{R}}^{n}$ be a rectangular region. Denote by $x=({x}_{1},{x}_{2},\cdots ,{x}_{n})$ a general point in ${\Omega}$ and write $dx=d{x}_{1}\cdots d{x}_{n}$. The main result embodied in Theorem 1 can be re-stated as follows.

Theorem 1. Let ${f}^{\alpha}$ $(\alpha =1,2,\cdots ,m)$ and their partial derivatives ${f}_{1}^{\alpha},{f}_{12}^{\alpha},\cdots ,{f}_{1\cdots (n-1)}^{\alpha}$, and ${\dot{f}}^{\alpha}:={f}_{1\cdots n}^{\alpha}$ are all defined and continuous on ${\Omega}$. Let further ${F}_{\alpha}:[0,\infty )\to [0,\infty )$ be any nonnegative differentiable functions, $\alpha =1,\cdots ,m$, with ${F}_{\alpha}^{\text{'}}$ nonnegative, continuous and nondecreasing on $[0,\infty )$. Suppose that

for all $x\in {\Omega}$, $\alpha =1,2,\cdots ,m$. 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.