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Let X = ∑k i=1 aiUi, Y = ∑k i=1 biUi, where the Ui are independent random vectors, each uniformly distributed on the unit sphere in R, and ai, bi are real constants. We prove that if {bi } is majorized by {ai } in the sense of Hardy–Littlewood–Pólya, and if Φ : R → R is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y ). Consequences include most of the known sharp L2-Lp 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 Ui are uniformly distributed on the unit ball of R instead of on the unit sphere. 1. Main results. In this first section, we set up notation and state our main results, about sums of independent random vectors uniformly distributed on spheres in R. History and background are provided in §2, Theorems 1, 2, 3 are proved in §3, 4, 5, and §6 contains a formula which permits transfer of the results in §1 to sums of independent random vectors uniformly distributed on balls in R. Throughout the paper, a = (a1, . . . , ak) and b = (b1, . . . , bk) will denote k-tuples of real numbers. Following the terminology of [MO 79], we say that a is majorized by b, and write a ≺ b, or {ai} ≺ {bi}, if

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