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

Let X = ∑_i=1^k a_iU_i, Y = ∑_i=1^k b_iU_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_i²} is majorized by {a_i² } 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.