A sharp integral Hardy type inequality and applications to Muckenhoupt weights on

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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 ϕ which satisfies the Muckenhoupt Aq condition with constant c upon all open subintervals of (0,1] should additionally satisfy the Aₚ 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.