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New generalizations of Hardy’s integral inequality. (English) Zbl 0944.26021
Consider the Hardy-type inequality $$ \int^b_a \Big(\int^x_a f(t) dt\Big)^p w(x) dx \le C \int^b_a f^p(x) v(x) dx\leqno(1) $$ on the class of all non-negative and measurable functions $f$ on $(a,b)$, $-\infty\le a<b\le +\infty$. 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 $0<a<b\le \infty$, $C=q^{p/r'}(r')^{-p/r'} [1-(\frac{a}{b})^{r'/q}]^{p/r'}$, $w(x) \equiv x^{-p/r'}$ and $v(x) \equiv 1$, where $p,r\in (1,\infty)$ and $p^{-1} + q^{-1} + r^{-1} = 1$.
Reviewer: B.Opic (Praha)

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
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