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

Albert Baernstein II, and Robert C. Culverhouse. "Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions." Studia Mathematica 152.3 (2002): 231-248. <http://eudml.org/doc/284723>.

@article{AlbertBaernsteinII2002,
abstract = {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.},
author = {Albert Baernstein II, Robert C. Culverhouse},
journal = {Studia Mathematica},
keywords = {sharp - Khinchin inequalities for sums; bisubharmonicity; majorization inequality},
language = {eng},
number = {3},
pages = {231-248},
title = {Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions},
url = {http://eudml.org/doc/284723},
volume = {152},
year = {2002},
}

TY - JOUR
AU - Albert Baernstein II
AU - Robert C. Culverhouse
TI - Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions
JO - Studia Mathematica
PY - 2002
VL - 152
IS - 3
SP - 231
EP - 248
AB - 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.
LA - eng
KW - sharp - Khinchin inequalities for sums; bisubharmonicity; majorization inequality
UR - http://eudml.org/doc/284723
ER -