Power means and the reverse Hölder inequality
Volume 207 / 2011
Studia Mathematica 207 (2011), 85-95
MSC: Primary 42B25, 42B35; Secondary 46E30.
DOI: 10.4064/sm207-1-6
Abstract
Let $w$ be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents $0<\alpha<\beta$. In the present paper, sharp estimates of the compositions of the power means $\mathcal{P}_\alpha w(x):=((1/x)\int_0^x w^\alpha(t)\,dt)^{1/\alpha}$, $x>0$, are obtained for various exponents $\alpha$. As a result, for the function $w$ a property of self-improvement of summability exponents is established.
