Multilinear direct and reverse Stolarsky inequalities

Document Type

Article

Publication Date

2002

Department

Department of Mathematical Sciences

Abstract

For any nonnegative measurable function f: [0, 1] → ℝ and any a > 0, let Q (f, a) denote the Stolarsky transform of f, equal to ∫01 f (x1/a) dx. Let Sn stand for the set of all permutations of the set {1, ..., n}. It is shown that the function (0, ∞)n ∋ a = (a1, ..., an) → script Q sign (a):= ∑σ∈Sn ∏i=1n Q (fσ(i),ai) is Schur-convex if the functions f1, ..., fn are nonnegative and nondecreasing and Schur-concave if f1, ..., fn are nonnegative and nonincreasing. Necessary and sufficient conditions for the strict Schur convexity and concavity are given. Similar results are obtained for certain "direct" and "reverse" extensions of the Stolarsky transform to measures.

Publication Title

Mathematical Inequalities and Applications

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