A note on a “Hardy-type” inequality. (English) Zbl 0703.26016
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.
Reviewer: A.Fiorenza
MSC:
26D15 | Inequalities for sums, series and integrals |