A note on a ``Hardy-type'' inequality (Q915893)

A generalization, involving certain continuous increasing functions, of the ``Hardy-type'' inequality \[ (*)\quad \frac{1}{\ell}\int^{\ell}_{0}(\frac{1}{t}\int^{t}_{0}f(s)ds)^{r/ q}dt\leq \frac{q}{q-r}(\frac{1}{\ell}\int^{\ell}_{0}f(s)ds)^{r/q}, \] (f\(\in L^ 1(0,\ell)\), \(f\geq 0\), \(1\leq r<q)\) is proved. The constant in (*) is the best possible.