for all
real (or
complex) numbers and where is the
cardinality of (the number of elements in ).
The inequality is named after the German mathematician
Hermann Minkowski.
Proof
First, we prove that has finite -norm if and both do, which follows by
Indeed, here we use the fact that is
convex over (for ) and so, by the definition of convexity,
This means that
Now, we can legitimately talk about If it is zero, then Minkowski's inequality holds. We now assume that is not zero. Using the triangle inequality and then
Hölder's inequality, we find that
We obtain Minkowski's inequality by multiplying both sides by
Minkowski's integral inequality
Suppose that and are two 𝜎-finite measure spaces and is measurable. Then Minkowski's integral inequality is:[1][2]
with obvious modifications in the case If and both sides are finite, then equality holds only if a.e. for some non-negative measurable functions and
If 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 for the integral inequality gives
If the measurable function is non-negative then for all [3]
This notation has been generalized to
for with Using this notation, manipulation of the exponents reveals that, if then
Reverse inequality
When the reverse inequality holds:
We further need the restriction that both and are non-negative, as we can see from the example and
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.
Generalizations to other functions
The Minkowski inequality can be generalized to other functions beyond the power function
The generalized inequality has the form
Various sufficient conditions on have been found by Mulholland[4] and others. For example, for one set of sufficient conditions from Mulholland is
Mahler's inequality – inequality relating geometric mean of two finite sequences of positive numbers to the sum of each separate geometric meanPages displaying wikidata descriptions as a fallback
^Mulholland, H.P. (1949). "On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality". Proceedings of the London Mathematical Society. s2-51 (1): 294–307.
doi:
10.1112/plms/s2-51.4.294.
for all
real (or
complex) numbers and where is the
cardinality of (the number of elements in ).
The inequality is named after the German mathematician
Hermann Minkowski.
Proof
First, we prove that has finite -norm if and both do, which follows by
Indeed, here we use the fact that is
convex over (for ) and so, by the definition of convexity,
This means that
Now, we can legitimately talk about If it is zero, then Minkowski's inequality holds. We now assume that is not zero. Using the triangle inequality and then
Hölder's inequality, we find that
We obtain Minkowski's inequality by multiplying both sides by
Minkowski's integral inequality
Suppose that and are two 𝜎-finite measure spaces and is measurable. Then Minkowski's integral inequality is:[1][2]
with obvious modifications in the case If and both sides are finite, then equality holds only if a.e. for some non-negative measurable functions and
If 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 for the integral inequality gives
If the measurable function is non-negative then for all [3]
This notation has been generalized to
for with Using this notation, manipulation of the exponents reveals that, if then
Reverse inequality
When the reverse inequality holds:
We further need the restriction that both and are non-negative, as we can see from the example and
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.
Generalizations to other functions
The Minkowski inequality can be generalized to other functions beyond the power function
The generalized inequality has the form
Various sufficient conditions on have been found by Mulholland[4] and others. For example, for one set of sufficient conditions from Mulholland is
Mahler's inequality – inequality relating geometric mean of two finite sequences of positive numbers to the sum of each separate geometric meanPages displaying wikidata descriptions as a fallback
^Mulholland, H.P. (1949). "On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality". Proceedings of the London Mathematical Society. s2-51 (1): 294–307.
doi:
10.1112/plms/s2-51.4.294.