Three-parameter weighted Hardy type inequalities

Abstract

For positive integer r and real numbers $1\leq p\leq q$ we find necessary and sufficient conditions for the validity of the following inequality: \begin{eqnarray*}\left(\int\limits_a^bu(x)\left(\int\limits_a^x|g(x)-g(t)|^rw(t)dt \right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}}\leq C\left( \int\limits_a^bv(x)|g'(x)|^pdx\right)^{\frac{1}{p}},\end{eqnarray*} where $u(\cdot)$, $v(\cdot)$, and $w(\cdot)$ are weight functions.