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

Albert Baernstein II¹ ; Robert C. Culverhouse²

¹Mathematics Department Washington University St. Louis, MO 63130, U.S.A.
²Department of Psychiatry Washington University Medical School St. Louis, MO 63110, U.S.A.

Studia Mathematica, Tome 152 (2002) no. 3, pp. 231-248

Résumé

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 ${{\mathbb R}}^n,$ and $a_i,b_i$ are real constants. We prove that if $\{ b_i^2\} $ is majorized by $\{ a_i^2\} $ in the sense of Hardy–Littlewood–Pólya, and if ${\mit \Phi }: {{\mathbb R}}^n \rightarrow {\mathbb R}$ is continuous and bisubharmonic, then $E{\mit \Phi }(X) \leq E{\mit \Phi }(Y)$. Consequences include most of the known sharp $L^2$-$L^p$ Khinchin inequalities for sums of the form $X.$ For radial ${\mit \Phi },$ 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 ${{\mathbb R}}^n$ instead of on the unit sphere.

Détail

DOI : 10.4064/sm152-3-3

Keywords: sum u sum u where independent random vectors each uniformly distributed unit sphere mathbb i real constants prove majorized sense hardy littlewood lya mit phi mathbb rightarrow mathbb continuous bisubharmonic mit phi leq mit phi consequences include known sharp l khinchin inequalities sums form radial mit phi bisubharmonicity necessary sufficient majorization inequality always counterparts majorization inequality exist uniformly distributed unit ball mathbb instead unit sphere