## On an inequality of G. Szegö.(English)Zbl 0734.26010

The following generalization of G. $$\{$$ not S. as in the References in this paper$$\}$$ 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 H.-T. Wang [Proc. Am. Math. Soc. 94, 641-646 (1985; Zbl 0583.26002)].

### MSC:

 26D15 Inequalities for sums, series and integrals 39B72 Systems of functional equations and inequalities 26B25 Convexity of real functions of several variables, generalizations

### Citations:

Zbl 0036.200; Zbl 0583.26002
Full Text:

### References:

 [1] Szegő, S, Über eine verallgemeinerung des dirichletschen integrals, Math. Z., 52, 676-685, (1950) · Zbl 0036.20001 [2] Mitrinović, D.S, Analytic inequalities, (1970), Berlin/Heidelberg/New York · Zbl 0199.38101 [3] Beckenbach, E.F; Bellman, R, Inequalities, (1971), Berlin/Heidelberg/New York · Zbl 0206.06802 [4] Karlin, S.J; Studden, W.J, Tchebycheff systems: with applications in analysis and statistics, (1966), New York/London/Sydney · Zbl 0153.38902 [5] Wang, H.T, Convex functions and Fourier coefficients, (), 641-646 · Zbl 0583.26002
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