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

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Baernstein A., Culverhouse R.

Studia Mathematica

2002

tom 152

nr 3

231-248

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.

Ograniczanie wyników

1 Studia Mathematica

1 Baernstein A.

1 Culverhouse R.

1 2002

Wyniki wyszukiwania

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