Karamata's inequality
E412925
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.
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.
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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 inequality
ⓘ
surface form:
Hardy–Littlewood–Pólya "Inequalities"
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
ⓘ
surface form:
Chebyshev's sum inequality
Hardy–Littlewood–Pólya inequality ⓘ Jensen inequality ⓘ
surface form:
Jensen's inequality
|
| hasAlternativeName |
Karamata's inequality
ⓘ
surface form:
Karamata majorization 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 inequality
ⓘ
surface form:
Hardy–Littlewood–Pólya theorem
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Karamata's inequality Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Jensen's inequality
this entity surface form:
Hardy–Littlewood–Pólya Inequalities
this entity surface form:
Karamata majorization inequality