Comparison theorems of isoperimetric type for moments of compact sets (Q1428934)

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 \(P_n(q)\).