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Refining the Hölder and Minkowski inequalities. (English) Zbl 0996.26011

In this paper, the author obtains and proves the refinements of the Hölder and Minkowski integral inequalities in the Lebesgue spaces \({\mathcal L}^p_\mu\). For the sake of completeness, we mention two of the important results obtained in this paper. Let \(p \geq 2\) and define \(p'\) by \(\frac 1{p} + \frac 1{p'} = 1\). Then for any two nonnegative \(\nu\)-measurable functions \(f\) and \(g\) \[ \int fg d\nu \leq \biggl(\int f^p d\nu - \int \biggl|f - g^{p'-1} \int fg d\nu \biggr/ \int g^{p'} d\nu \biggr|^p d\nu\biggr)^{1/p} \biggl(\int g^{p'} d\nu\biggr)^{1/p'}\tag{1} \] and \[ \biggl(\int (f+g)^p d\nu \biggr)^{1/p} \leq \biggl(\int f^p d\nu - \int h^p d\nu\biggr)^{1/p} + \biggl(\int g^p d\nu - \int h^p d\nu\biggr)^{1/p},\tag{2} \] where \[ h = \biggl|f\int g(f+g)^{p-1} d\nu- g\int f(f+g)^{p-1} d\nu\biggr|\biggl/ \int (f+g)^p d\nu. \] Inequalities (1) and (2) are the refined Hölder and Minkowski integral inequalities, respectively. Furthermore, the lower bounds for the above inequalities are obtained for the case \(1 < p \leq 2\). Interestingly, when \(p = 2\), both inequalities reduce to equalities in \({\mathcal L}^2_\mu\). Examples are given to illustrate the fact that the nonnegativity assumptions on \(f\) and \(g\) cannot be dropped.

MSC:

26D15 Inequalities for sums, series and integrals
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