The reverse Holder inequality for an elementary function

Abstract

For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by

\smallskip\centerline{$\displaystyle
\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),
\qquad
\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$}

Assume that $0<A<B$, $0<\theta<1$ and consider the step function
$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$, where $\chi_E$ is the characteristic function of the set $E$.

Let $-\infty<p<q<+\infty$. The main result of this work consists in finding the term

\smallskip\centerline{$\displaystyle
C_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$}

\smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\theta_{p<q,A<B}$ with respect to $\beta=B/A\in(1,+\infty)$.
The cases $p=0$ or $q=0$ are considered separately.

The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].