## 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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### References:

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