Hardy type inequality with sharp constant for $0 < p < 1$

A power-weighted integral inequality with sharp constant for $0 < p < 1$ was established by V.I. Burenkov for the Hardy operator $(Hf)(x)=\frac1x\int_0^xf(t)\,dt$ for non-negative non-increasing functions $f$. In this work we consider a more general class of functions $f$ and prove a new Hardy-type inequality with sharp constant for functions of this class.