Some inequalities for products of power sums

Bruce Reznick

Vol. 104 (1983), No. 2, 443–463

DOI: 10.2140/pjm.1983.104.443

Abstract

We study the asymptotic behavior of the range of the ratio of products of power sums. For x = (x₁,…,x_n), define M_p = M_p(x) = ∑ x_i^p. As two representative and explicit results, we show that the maximum and minimum of the function M₁M₃∕M₂² are ±3∕16n^1∕2 + 5∕8 + 𝒪(n^−1∕2) 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 M_p1^a₁⋯M_{p r}^a_r∕M _q^b, ∑a _ip_i = 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