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Some integral inequalities of Hölder and Minkowski type. (English) Zbl 1122.26022
The usual Hölder and Minkowski integral inequality is of the following form
$\int_a^b fg \,dx \leq \left(\int_a^b f^\alpha \,dx \right)^{1/\alpha}\left(\int_a^b g^\beta\, dx \right)^{1/\beta}$ and
$\left(\int_a^b (f+g)^\alpha \,dx \right)^{1/\alpha} \leq \left(\int_a^b f^\alpha \,dx \right)^{1/\alpha}+\left(\int_a^b g^\alpha \,dx \right)^{1/\alpha},$ respectively, where $$f, g : [a,b] \to [0,\infty]$$ be nonnegative functions and $$1<\alpha<\infty, \alpha^{-1}+\beta^{-1}=1$$.
The author establishes some integral inequalities for a class of generalized weighted quasi-arithmetic mean, i.e. some analogue of the classical Hölder and Minkowski integral inequalities. Some well-known inequalities and corresponding analogues are obtained from those new Hölder and Minkowski integral inequalities.

##### MSC:
 26D15 Inequalities for sums, series and integrals
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