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 whose proof is presented in this paper. As an application we find the exact best possible range of p > q such that any non-increasing g which satisfies a reverse Holder inequality with exponent q and constant c upon the subintervals of (0,1], should additionally satisfy a reverse Holder inequality with exponent p and in general a different constant The result has been treated in [1] but here we give an alternative proof based on the above mentioned inequality.