On an inequality of G. Szegö (Q810650)

The following generalization of \textit{G.} \(\{\) not \textit{S.} as in the References in this paper\(\}\) \textit{Szegö}'s inequality [Math. Z. 52, 676-685 (1950; Zbl 0036.200)] is offered. Let I be an interval in \({\mathbb{R}}^ n\), partially ordered by \(x=(x_ 1,...,x_ n)\leq y=(y_ 1,...,y_ n)\) iff \(x_ k\leq y_ k\quad (k=1,...,n)\) and \(f: I\to {\mathbb{R}}\) satisfy \(f(x+h)-f(x)\leq f(y+h)-f(y)\) whenever \(x,y+h\in I,\) \(x\leq y,\) \(0\leq h\in {\mathbb{R}}^ n.\) Then, for \(t_ j\in I\quad (j=1,...,2m+1),\) \(t_ 1\geq t_ 2\geq...\geq t_{2m+1},\) \[ \sum^{m+1}_{k=1}(-1)^{k-1}f(t_ k)\geq f(\sum^{2m+1}_{k=1}(- 1)^{k-1}t_ k). \] This is used for a simpler proof of an inequality of \textit{H.-T. Wang} [Proc. Am. Math. Soc. 94, 641-646 (1985; Zbl 0583.26002)].