A SHARP INTEGRAL HARDY TYPE INEQUALITY AND APPLICATIONS TO MUCKENHOUPT WEIGHTS ON R

We prove a generalization of a Hardy type inequality for negative exponents valid for non-negative functions defined on [0, 1). As an application we find the exact best possible range of p such that 1 < p <= q such that any non-decreasing phi which satisfies the Muckenhoupt A(q) condition with constant c upon all open subintervals of [0, 1) should additionally satisfy the A(p) condition for another possibly real constant c'. The result have been treated in [9] based on [1], but we give in this paper an alternative proof which relies on the above mentioned inequality.