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On new generalizations of Hardy’s integral inequality. (English) Zbl 0893.26008

The well-known Hardy’s inequality states that

${\int }_{0}^{\infty }{\left[{x}^{-1}{\int }_{0}^{x}f\left(t\right)dt\right]}^{p}dx<{q}^{p}{\int }_{0}^{\infty }{f}^{p}\left(t\right)dt\phantom{\rule{2.em}{0ex}}\left(1\right)$

provided that $p\in \left(1,\infty \right)$, $q=p/\left(p-1\right)$, $f>0$ and $f¬\equiv 0$. The inequality (1) implies that for any $a$ and $b$ satisfying $0,

${\int }_{a}^{b}{\left[{x}^{-1}{\int }_{a}^{x}f\left(t\right)dt\right]}^{p}<{q}^{p}{\int }_{a}^{b}{f}^{p}\left(t\right)dt,\phantom{\rule{1.em}{0ex}}f>0,\phantom{\rule{1.em}{0ex}}f¬\equiv 0·\phantom{\rule{2.em}{0ex}}\left(2\right)$

One of the main results of the paper shows that the constant ${q}^{p}$ in (2) can be replaced by ${q}^{p}{\left[1-{\left(a/b\right)}^{1/q}\right]}^{p}$. The rest of the paper represents other variants of (1).

Reviewer: B.Opic (Praha)

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions
##### Keywords:
Hardy inequality; integral inequality