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We prove a sharp integral inequality valid for non-negative functions definedon $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the wellknown integral Hardy inequality. We prove it as a consequence of the respectiveweighted 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 anynon-increasing $f$ which satisfies a reverse H\"{o}lder inequality withexponent $q$ and constant $c$ upon the subintervals of $[0,1]$, shouldadditionally satisfy a reverse H\"{o}lder inequality with exponent $p$ and adifferent in general constant $c'$. The result has been treated in \cite{1} buthere we give an alternative proof based on the above mentioned inequality.

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