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

Tom 152 / 2002

Albert Baernstein II, Robert C. Culverhouse Studia Mathematica 152 (2002), 231-248 MSC: Primary 60E15; Secondary 60G50. DOI: 10.4064/sm152-3-3


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.


  • Albert Baernstein IIMathematics Department
    Washington University
    St. Louis, MO 63130, U.S.A.
  • Robert C. CulverhouseDepartment of Psychiatry
    Washington University Medical School
    St. Louis, MO 63110, U.S.A.

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