On some reverse integral inequalities. (English) Zbl 0910.26010
Let \(a\in (0,\infty)\) be fixed. For \(\gamma, q\in(1,\infty)\) define \(M(\gamma, q)\) as the set of all positive functions on \((0,a]\) satisfying
\[
f^q(t)\leq t^{-1} \int^t_0 f^q(x)dx\leq \gamma f^q(t),\quad t\in(0, a].
\]
The main result of the paper states that \(M(\gamma, q)\subset L^p(0,a)\) for any \(p\in [q,q\gamma/(\gamma- 1))\) and that for all \(f\in M(\gamma, q)\),
\[
\Biggl({1\over a} \int^a_0 f^p(x)dx\Biggr)^{1/p}\leq c\Biggl({1\over a} \int^a_0 f^q(x)dx\Biggr)^{1/q},
\]
where \(c= [\gamma^{r+ 1}/(\gamma- r(\gamma- 1))]^{1/p}\) and \(r= p/q\).
Reviewer: B.Opic (Praha)
MSC:
26D15 | Inequalities for sums, series and integrals |
Keywords:
reverse Hölder inequalityReferences:
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