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 $1<p\le q$ such that any non-decreasing $ϕ$ 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 \cite{9} based on \cite{1}, but we give in this paper an alternative proof which relies on the above mentioned inequality.

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