Some generalized Opial-type inequalities (Q1190569)

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 \(x=(x_ 1,x_ 2,\dots,x_ n)\) a general point in \(\Omega\) and write \(dx=dx_ 1\dots dx_ n\). The main result embodied in Theorem 1 can be re-stated as follows. Theorem 1. Let \(f^ \alpha\) \((\alpha=1,2,\dots,m)\) and their partial derivatives \(f^ \alpha_ 1,f^ \alpha_{12},\dots,f^ \alpha_{1\dots(n-1)}\), and \(\dot f^ \alpha:=f^ \alpha_{1\dots n}\) are all defined and continuous on \(\Omega\). Let further \(F_ \alpha:[0,\infty)\to[0,\infty)\) be any nonnegative differentiable functions, \(\alpha=1,\dots,m\), with \(F_ \alpha'\) nonnegative, continuous and nondecreasing on \([0,\infty)\). Suppose that \[ f^ \alpha(a_ 1,x_ 2,\dots,x_ n)=f^ \alpha_ 1(x_ 1,a_ 2,x_ 3,\dots,x_ n)=\cdots=f^ \alpha _{1\dots(n-1)}(x_ 1,\dots,x_{n- 1},a_ n)=0 \] for all \(x\in\Omega\), \(\alpha=1,2,\dots,m\). Then we have \[ \int_ \Omega\sum^ m_{\beta=1}\left(\prod_{\alpha\neq\beta}F_ \alpha(| f^ \alpha(x)|)\right)F_ \beta'(| f^ \beta(x)|)|\dot f^ \beta(x)| dx\leq\prod^ m_{\alpha=1}F_ \alpha\bigl(\int_ \Omega|\dot f^ \alpha(x)| dx\bigr). \] Another inequality given in Theorem 2 is obtained under the additional condition \[ f^ \alpha(b_ 1,x_ 2,\dots,x_ n)=f^ \alpha_ 1(x_ 1,b_ 2,x_ 3,\dots,x_ n)=\cdots=f^ \alpha _{1\dots(n-1)}(x_ 1,\dots,x_{n-1},b_ n)=0. \] Some known inequalities due to \textit{G. S. Yang} [Tamkang J. Math. 13, 255-259 (1982; Zbl 0516.26009)] established for two-variable functions are contained in the results obtained.