The results for higher moments follow from Hölder's inequality, which implies that higher moments ( or halves of moments ) diverge if lower ones do.

It has a number of generalizations, among them Hölder's inequality.

* Hölder's inequality

In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of L < sup > p </ sup > spaces.

For 1 < p, q < ∞ and f ∈ L < sup > p </ sup >( μ ) and g ∈ L < sup > q </ sup >( μ ), Hölder's inequality becomes an equality if and only if | f |< sup > p </ sup > and | g |< sup > q </ sup > are linearly dependent in L < sup > 1 </ sup >( μ ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α | f |< sup > p </ sup > = β | g |< sup > q </ sup > μ-almost everywhere.

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L < sup > p </ sup >( μ ), and also to establish that L < sup > q </ sup >( μ ) is the dual space of L < sup > p </ sup >( μ ) for 1 ≤ p < ∞.

Hölder's inequality was first found by, and discovered independently by.

The brief statement of Hölder's inequality uses some conventions.

* On the right-hand side of Hölder's inequality, 0 times ∞ as well as ∞ times 0 means 0.

and the similar one for fg hold, and Hölder's inequality can be applied to the right-hand side.

In particular, if f and g are in the Hilbert space L < sup > 2 </ sup >( μ ), then Hölder's inequality for p = q = 2 implies

A combination of Hölder's inequality and Jensen's inequality implies that

# REDIRECT Hölder's inequality

# REDIRECT Hölder's inequality

He is famous for many things including: Hölder's inequality, the Jordan – Hölder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive group of real numbers, the classification of simple groups of order up to 200, and Hölder's theorem which implies that the Gamma function satisfies no algebraic differential equation.

# REDIRECT Hölder's inequality

# REDIRECT Hölder's inequality

# REDIRECT Hölder's inequality

This result, known as Jensen's inequality underlies many important inequalities ( including, for instance, the arithmetic-geometric mean inequality and Hölder's inequality ).

More generally, by Hölder's inequality, it follows that if ƒ ∈ L < sup > p </ sup >( a, b ), then I < sup > α </ sup > ƒ ∈ L < sup > p </ sup >( a, b ) as well, and the analogous inequality holds

Then Hahn's Embedding Theorem reduces to Hölder's theorem ( which states that a linearly ordered abelian group is Archimedean if and only if it is a subgroup of the ordered additive group of the real numbers ).

in view of Hölder's inequality, that S is bounded from to L < sup >∞</ sup >, where again.

This form of Young's inequality is a special case of the inequality of weighted arithmetic and geometric means and can be used to prove Hölder's inequality.

The latter formulation follows from the former through an application of Hölder's inequality and a duality argument.

Young's inequality for products can be used to prove Hölder's inequality.

The triangle inequality holds.

In every polygon with perimeter p and area A, the isoperimetric inequality holds.

< li > If is a symmetric matrix of the form, and the strict inequality holds

* For a differentiable Lipschitz map ƒ: U → R < sup > m </ sup > the inequality holds for the best Lipschitz constant of f, and it turns out to be an equality if the domain U is convex.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment ( that is, one with central angle in π ) with those endpoints.

In mathematics, an inequation is a statement that an inequality holds between two values.

In mathematics, an inequality is a relation that holds between two values when they are different ( see also: equality ).

To bring the numerical values of parameters μ, σ into the domain where strong inequality holds true one could use the fact that if X is log-normally distributed then X < sup > m </ sup > is also log-normally distributed with parameters μm, σm.

Using the above definition of ARA, the following inequality holds for DARA:

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

In order to " measure the size " of A, it then seems natural to take the smallest number c such that the above inequality holds for all v in V. In other words, we measure the " size " of A by how much it " lengthens " vectors in the " biggest " case.

For instance, the function log ( x ) is concave ( note that we can use Jensen's to prove convexity or concavity, if it holds for two real numbers whose functions are taken ), so substituting in the previous formula ( 4 ) establishes the ( logarithm of ) the familiar arithmetic mean-geometric mean inequality:

The following year he took up a lectureship in the University of Bordeaux, where he proved his celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds.

Although a discussion exists about the recent trends in global inequality, the issue is anything but clear, and this holds true for both the overall global inequality trend and for its between-country and within-country components.

* Jensen's inequality holds: If ƒ is a convex function, then

Distance measures are often classified into Euclidean measures and non-Euclidean measures depending on whether the triangle inequality holds.

Fubini's theorem also shows that this operator is continuous with respect to the Banach space structure on L < sup > 1 </ sup >, and that the following inequality holds:

Hence the inequality holds.

In addition an inequality for the norms holds, namely

Then the following interpolation inequality holds for all r between p and q and all f ∈ L < sup > r </ sup >:

WSA holds that struggles against gender inequality, structural racism, and oppression of LGBTQ people are also part of the larger fight for social liberation and self-management.