Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

Albert Baernstein, Robert C. Culverhouse

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

Let X = ∑i=1k aiUi, Y = ∑i=1k biUi, where the Ui are independent random vectors, each uniformly distributed on the unit sphere in ℝn, and ai, bi are real constants. We prove that if {bi2} is majorized by {ai2 } in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝn → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp L2-Lp Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts to the majorization inequality exist when the Ui are uniformly distributed on the unit ball of ℝn instead of on the unit sphere.

Original languageEnglish
Pages (from-to)231-248
Number of pages18
JournalStudia Mathematica
Volume152
Issue number3
DOIs
StatePublished - Jan 1 2002

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