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Some generalized Opial-type inequalities. (English) Zbl 0761.26012

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

Let ${\Omega }:=\left[{a}_{1},{b}_{1}\right]×\left[{a}_{2},{b}_{2}\right]×\cdots ×\left[{a}_{n},{b}_{n}\right]\subset {ℝ}^{n}$ be a rectangular region. Denote by $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$ 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 ${f}^{\alpha }$ $\left(\alpha =1,2,\cdots ,m\right)$ and their partial derivatives ${f}_{1}^{\alpha },{f}_{12}^{\alpha },\cdots ,{f}_{1\cdots \left(n-1\right)}^{\alpha }$, and ${\stackrel{˙}{f}}^{\alpha }:={f}_{1\cdots n}^{\alpha }$ are all defined and continuous on ${\Omega }$. Let further ${F}_{\alpha }:\left[0,\infty \right)\to \left[0,\infty \right)$ be any nonnegative differentiable functions, $\alpha =1,\cdots ,m$, with ${F}_{\alpha }^{\text{'}}$ nonnegative, continuous and nondecreasing on $\left[0,\infty \right)$. Suppose that

${f}^{\alpha }\left({a}_{1},{x}_{2},\cdots ,{x}_{n}\right)={f}_{1}^{\alpha }\left({x}_{1},{a}_{2},{x}_{3},\cdots ,{x}_{n}\right)=\cdots ={f}_{1\cdots \left(n-1\right)}^{\alpha }\left({x}_{1},\cdots ,{x}_{n-1},{a}_{n}\right)=0$

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

${\int }_{{\Omega }}\sum _{\beta =1}^{m}\left(\prod _{\alpha \ne \beta }{F}_{\alpha }\left(|{f}^{\alpha }\left(x\right)|\right)\right){F}_{\beta }^{\text{'}}\left(|{f}^{\beta }\left(x\right)|\right)|{\stackrel{˙}{f}}^{\beta }\left(x\right)|dx\le \prod _{\alpha =1}^{m}{F}_{\alpha }\left({\int }_{{\Omega }}|{\stackrel{˙}{f}}^{\alpha }\left(x\right)|dx\right)·$

Another inequality given in Theorem 2 is obtained under the additional condition

${f}^{\alpha }\left({b}_{1},{x}_{2},\cdots ,{x}_{n}\right)={f}_{1}^{\alpha }\left({x}_{1},{b}_{2},{x}_{3},\cdots ,{x}_{n}\right)=\cdots ={f}_{1\cdots \left(n-1\right)}^{\alpha }\left({x}_{1},\cdots ,{x}_{n-1},{b}_{n}\right)=0·$

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.

##### MSC:
 26D10 Inequalities involving derivatives, differential and integral operators 26D15 Inequalities for sums, series and integrals of real functions