Comparison theorems of isoperimetric type for moments of compact sets
type: Item
instance of: scholarly article
title: Comparison theorems of isoperimetric type for moments of compact sets
zbMATH Open document ID: 1074.26006
author: D. Massart
published in: Collectanea Mathematica
publication date: 2004-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/44324
review text: The authors consider \(q\)-moments of the form \[ I_n(q)=\int_\Omega| x|^{q-n}f(x)\,dx \quad (q>0), \] where \(\Omega\subset{\mathbb R}^n\) is compact, and Stieltjes integrals \[ P_n(q)=\int_0^{x_1} d\psi_1^q(y_1) \int_0^{x_2}d\psi_2^q(y_2)\cdots\int_0^{x_n}h(y) \varphi^q(y)\,d\psi_n^q(y_n), \] where \(x_k>0\) \((k=1,2,\dots,n)\), \(y=(y_1,y_2,\dots,y_n)\) and \(\psi_k\) are absolutely continuous and strictly increasing, with \(\psi_k(0)=0\), for all \(k\). The aim of the paper is to compare \(I_n(q_1)\) with \(I_n(q_2)\), and \(P_n(q_1)\) with \(P_n(q_2)\) when \(0<q_1<q_2<\infty.\) The authors present a new unified approach to prove some inequalities, generalizing results due to G. Pólya and C. Szegő (the isoperimetric inequalities) for the expressions \(I_n(q)\), and to E.M. Stein (in the context of interpolating spaces \(L(p,q)\)), for the expressions \(P_n(q)\).
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A42
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 2065908
zbMATH Keywords: moments
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: Stieltjes integral
zbMATH Keywords: multinomial distribution
reviewed by: Florian-Horia Vasilescu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2065908