Hardy inequalities for $p$-weakly monotone functions

We prove Hardy-type inequalities
$$\left(\int_d^\infty\left|\int_d^s f(x)dx\right|^p s^\beta ds\right)^{1/p}\leqslant C\left(\int_d^\infty|f(s)|^qs^\alpha ds\right)^{1/q}$$
for the class of $p$-weakly monotone functions with $q$ or $p$ smaller than $1$ and $d\geqslant 0$.