A Hardy inequality and applications to reverse Holder inequalities for weights on $R$

We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective weighted discrete analogue inequality which proof is presented in this paper. As an application we find the exact best possible range of $p>q$ such that any non-increasing $f$ which satisfies a reverse H\"{o}lder inequality with exponent $q$ and constant $c$ upon the subintervals of $[0,1]$, should additionally satisfy a reverse H\"{o}lder inequality with exponent $p$ and a different in general constant $c'$. The result has been treated in \cite{1} but here we give an alternative proof based on the above mentioned inequality.