Minkowski's inequality

Minkowski's inequality

[miŋ′kȯf·skēz ‚in·i′kwäl·əd·ē]
(mathematics)
An inequality involving powers of sums of sequences of real or complex numbers, ak and bk : provided s ≥ 1.
An inequality involving powers of integrals of real or complex functions, ƒ and g, over an interval or region R : provided s ≥ 1 and the integrals involved exist.
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They begin by describing 14 classical and new inequalities, among them inequalities between means, general weighted power mean inequalities, Minkowski's inequality, the rearrangement inequality, and Popoviciu's inequality.
Then for any measurable set E c R, using Minkowski's inequality, we have the estimate
By the definition and Minkowski's inequality, we easily deduce that
Also, Holder's inequality is used to prove Minkowski's inequality (the triangle inequality for [L.sub.p] spaces) and to establish that [L.sub.q]([mu]) is the dual space of [L.sub.p]([mu]) for p [member of] [1, [infinity]).
Using an easy change of variables and Minkowski's inequality, we obtain
Then by Minkowski's inequality for integrals, the semigroups estimates (see e.g.
First let us recall the well known Minkowski's inequality. For details see [9].
Liu, Minkowski's inequality for extended mean values, Proceedings of the Second ISAAC Congress, Int.
Since Y1 is bounded, in order to prove our theorem, it is sufficient, by Minkowski's inequality, to show that
By Minkowski's inequality for k > 1, to complete the proof of Lemma 3.3, it is sufficient to
We obtain the following bounds using Lemma 5.2 and Minkowski's inequality.