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

Albert Baernstein II ; Robert C. Culverhouse

Studia Mathematica, Tome 151 (2002), p. 231-248 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $X = ∑_{i=1}^{k} a_{i}U_{i}$, $Y = ∑_{i=1}^{k} b_{i}U_{i}$, 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 ${b²_{i}}$ is majorized by ${a²_{i}}$ 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 $L²-L^{p}$ 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 to the majorization inequality exist when the $U_{i}$ are uniformly distributed on the unit ball of ℝⁿ instead of on the unit sphere.

Détail

Publié le : 2002-01-01

Zbl 0997.60009

EUDML-ID : urn:eudml:doc:284723