Vol. 104, No. 2, 1983

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Some inequalities for products of power sums

Bruce Reznick

Vol. 104 (1983), No. 2, 443–463
Abstract

We study the asymptotic behavior of the range of the ratio of products of power sums. For x = (x1,,xn), define Mp = Mp(x) = xip. As two representative and explicit results, we show that the maximum and minimum of the function M1M3∕M22 are ±3√316n12 + 58 + 𝒪(n12) and that n M1M3∕M4 > n∕8, where “1/8” is the best possible constant. We give readily computable, if less explicit, formulas of this kind for Mp1a1Mp rar∕M qb, a ipi = bq. Applications to integral inequalities are discussed. Our results generalize the classical Hölder and Jensen inequalities. All proofs are elementary.

Mathematical Subject Classification 2000
Primary: 26D15
Secondary: 26B25, 26C05, 10C25
Milestones
Received: 10 March 1981
Published: 1 February 1983
Authors
Bruce Reznick
Department of Mathematics and Center for Advanced Study
University of Illinois at Urbana-Champaign
1409 W. Green Street
327 Altgeld Hall
Urbana IL 61801-2975
United States