A general version of Young's inequality (Q811673)

In this note the classical Young inequality is generalized: Let \(F: I\to\mathbb{R}\) be a continuous and continuously differentiable function on \(I\). If \(n\geq 2\) then assume that \(x_ j\mapsto\partial_ iF(x_ 1,\dots,x_ n)\), \(x_ j\in[a_ j,b_ j]\), is a nondecreasing function for all \(i,j=1,\dots,n\), \(i\neq j\). Let further \(\varphi: [\alpha,\beta]\to I\) be a monotonic curve connecting \(a\) and \(b\), i.e. \(\varphi(\alpha)=a\), \(\varphi(\beta)=b\). Then \[ F(u)- F(a)\leq\sum_{i=1}^ n(\varphi)\int_{a_ i}^{u_ i}\partial_ iF(x)dx_ i \] holds for all \(u=(u_ 1,\dots,u_ n)\in I\). (Here \((\varphi)\int_{a_ i}^{u_ i}\partial_ i F(x)dx_ i\) is the integral of \(f\) along the curve \(\varphi\) between the hyperplanes \(x_ i=a_ i\) and \(x_ i=u_ i\) in the direction of the \(i\)-th coordinate axis.) Taking \(n=2\) and \(F(x,y)=xy\), one obtains the classical Young inequality. This result generalizes an earlier result of the author [Acta Sci. Math. 54, No. 3/4, 327-338 (1990; Zbl 0732.39007)].