Weighted integral inequalities with the geometric mean operator
The geometric mean operator is defined by \(Gf(x)=\exp\left( \frac{1}{x} \int_{0}^{x}\log f(t)dt\right) \) for locally integrable \(f\geq 0.\) Given two weights \(u,v\geq 0\) on \((0,\infty)\) and two indices \(0<p,q<\infty,\) one associates to them the function \(w=\left( G(1/v)\right) ^{q/p}u.\) The paper under review contains two-sided estimates (in both cases \(p\leq q\) and \(q<p\)) of the best constant \(C\) such that \[ \left( \int_{0}^{\infty}(Gf)^{q}udx\right) ^{1/q}\leq C\left( \int _{0}^{\infty}f^{p}v dx\right) ^{1/p} \] for all \(f\in L^{p}(vdx),\) \(f\geq 0.\) For example, if \(p\leq q\) and \[ D=\sup_{t>0}t^{-1/p}\left( \int_{0}^{t}w(x)dx\right) ^{1/q}<\infty, \] then a best constant \(C\) exists and \(D\leq C\leq e^{1/p}D.\) The factor \(e^{1/p}\) is best possible for \(p=q\) and attains for \(u=v=1.\)
scientific article; zbMATH DE number 1910309