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On inverses of the Hölder inequality. (English) Zbl 0752.26009

The author derives an inequality complementary to \[ x^{1/p} y^{1/q}\leq x/p+y/q. \] Using this inequality, then converse Hölder- type inequalities of the form \[ a\| f\|_ 1+b\| g\|_ 1\leq c\| f^{1/p} g^{1/q}\|_ 1\qquad\text{and}\qquad \| f\|_ p\| g\|_ q\leq d\| fg\|_ 1 \] are obtained, where the ranges of \(f\) and \(g\) are compact subintervals of \((0,\infty)\). The constants \(c\) and \(d\) then depend on \(a\), \(b\), \(p\), \(q\) and the endpoints of the intervals containing the ranges of \(f\) and \(g\).
The results include a lot of earlier inequalities complementary to Hölder’s one.

MSC:

26D15 Inequalities for sums, series and integrals
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