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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
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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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