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We prove a generalization of a Hardy type inequality for negative exponentsvalid for non-negative functions defined on $(0,1]$. As an application we findthe exact best possible range of $p$ such that $1<p\le q$ such that anynon-decreasing $\phi$ which satisfies the Muckenhoupt $A_q$ condition withconstant $c$ upon all open subintervals of $(0,1]$ should additionally satisfythe $A_p$ condition for another possibly real constant $c'$. The result havebeen treated in \cite{9} based on \cite{1}, but we give in this paper analternative proof which relies on the above mentioned inequality.

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