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.
Then the discrete and integral forms of
Minkowski's inequality are given as
