Refining the Hölder and Minkowski inequalities (Q1347686)

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.