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

Albert Baernstein II; Robert C. Culverhouse — 2002

Studia Mathematica

Let $X = \sum_{i = 1}^{k} a_{i} U_{i}$ , $Y = \sum_{i = 1}^{k} b_{i} U_{i}$ , where the $U_{i}$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_{i}, b_{i}$ are real constants. We prove that if $b ²_{i}$ is majorized by $a ²_{i}$ 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 $L ² - L^{p}$ 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...

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