Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces.

are positively linearly dependent; that is,

In fact, it is a special case of the more general fact

where it is easy to see that the right-hand side satisfies the triangular inequality.

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

for all real (or complex) numbers

In probabilistic terms, given the probability space

denote the expectation operator for every real- or complex-valued random variables

Minkowski's inequality reads

The inequality is named after the German mathematician Hermann Minkowski.

If it is zero, then Minkowski's inequality holds.

Using the triangle inequality and then Hölder's inequality, we find that

We obtain Minkowski's inequality by multiplying both sides by

, one has, by convexity, By integration this leads to One takes then to reach the conclusion.

are two 𝜎-finite measure spaces and

Then Minkowski's integral inequality is:[1][2]

with obvious modifications in the case

and both sides are finite, then equality holds only if

for some non-negative measurable functions

is the counting measure on a two-point set

then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting

Using this notation, manipulation of the exponents reveals that, if

the reverse inequality holds:

The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range.

Using the Reverse Minkowski, we may prove that power means with

such as the harmonic mean and the geometric mean are concave.

The Minkowski inequality can be generalized to other functions

beyond the power function

The generalized inequality has the form

one set of sufficient conditions from Mulholland is