Karamata's inequality

E412925

mathematical inequality result in majorization theory

Karamata's inequality is a fundamental result in majorization theory that generalizes several classical inequalities by comparing sums of convex (or concave) functions over majorized sequences.

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All labels observed (3)

Label	Occurrences
Hardy–Littlewood–Pólya Inequalities	1
Karamata majorization inequality	1
Karamata's inequality canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T4092204 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Karamata's inequality
Context triple: [Jensen's inequality, relatedTo, Karamata's inequality]

A. Jensen inequality
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
B. Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
C. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
D. Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
E. Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Karamata's inequality
Target entity description: Karamata's inequality is a fundamental result in majorization theory that generalizes several classical inequalities by comparing sums of convex (or concave) functions over majorized sequences.

A. Jensen inequality
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
B. Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
C. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
D. Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
E. Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical inequality ⓘ result in majorization theory ⓘ
appearsIn	Hardy–Littlewood–Pólya "Inequalities" ⓘ linked to: Hardy–Littlewood–Pólya inequality texts on majorization and matrix inequalities ⓘ
appliesTo	real sequences ⓘ vectors in R^n ⓘ
assumes	one sequence majorizes another ⓘ
characterizes	Schur-concave functions ⓘ Schur-convex functions ⓘ
conclusion	sum of concave function over majorizing sequence is less or equal ⓘ sum of convex function over majorizing sequence is greater or equal ⓘ
domainCondition	finite sequences of real numbers ⓘ sequences sorted in decreasing order ⓘ
field	inequality theory ⓘ majorization theory ⓘ mathematical analysis ⓘ
generalizes	Chebyshev's sum inequality ⓘ linked to: Chebyshev’s sum inequality Hardy–Littlewood–Pólya inequality ⓘ Jensen's inequality ⓘ linked to: Jensen inequality
hasAlternativeName	Karamata majorization inequality ⓘ linked to: Karamata's inequality
hasFormulation	if x majorizes y and f is concave then Σ f(x_i) ≤ Σ f(y_i) ⓘ if x majorizes y and f is convex then Σ f(x_i) ≥ Σ f(y_i) ⓘ
hasProperty	order-preserving for convex sums under majorization ⓘ order-reversing for concave sums under majorization ⓘ
holdsFor	strictly concave functions with strict reversed inequality when sequences differ nontrivially ⓘ strictly convex functions with strict inequality when sequences differ nontrivially ⓘ
implies	comparison of power means ⓘ monotonicity of Schur-convex functions under majorization ⓘ
isToolFor	comparing distributions of resources ⓘ studying dispersion of sequences ⓘ
namedAfter	Jovan Karamata ⓘ
relatedTo	Hardy–Littlewood–Pólya theorem ⓘ linked to: Hardy–Littlewood–Pólya inequality majorization order ⓘ rearrangement inequalities ⓘ
requires	convex function defined on interval containing sequences ⓘ
timePeriod	20th century mathematics ⓘ
usedIn	economics of inequality ⓘ information theory ⓘ matrix analysis ⓘ optimization theory ⓘ probability theory ⓘ proofs of classical inequalities ⓘ
usedToDerive	Muirhead's inequality ⓘ inequalities between symmetric means ⓘ
usesConcept	concave function ⓘ convex function ⓘ majorization ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jensen's inequality → relatedTo → Karamata's inequality ⓘ

subject linked to: Jensen inequality

Inequalities → hasAbbreviation → Hardy–Littlewood–Pólya Inequalities ⓘ

linked to: Karamata's inequality

Karamata's inequality → hasAlternativeName → Karamata majorization inequality ⓘ