New generalizations of Hardy's integral inequality (Q1970952)

Consider the Hardy-type inequality \[ \int^b_a \Big(\int^x_a f(t) dt\Big)^p w(x) dx \leq C \int^b_a f^p(x) v(x) dx\leqno(1) \] on the class of all non-negative and measurable functions \(f\) on \((a,b)\), \(-\infty\leq a<b\leq +\infty\). If \(s \in (1,\infty)\), put \(s' = s/(s-1)\). The classical Hardy inequality states that (1) holds with \((a,b) = (0,\infty)\), \(C=(p')^p\), \(w(x)\equiv x^{-p}\) and \(v(x) \equiv 1\), where \(p\in (1,\infty)\). The authors of the paper under review prove three generalizations of the classical Hardy inequality. To illustrate their results, we mention one of them (Theorem ~2.1): The inequality (1) holds if \(0<a<b\leq \infty\), \(C=q^{p/r'}(r')^{-p/r'} [1-(\frac{a}{b})^{r'/q}]^{p/r'}\), \(w(x) \equiv x^{-p/r'}\) and \(v(x) \equiv 1\), where \(p,r\in (1,\infty)\) and \(p^{-1} + q^{-1} + r^{-1} = 1\).