Young's inequality
E412926
functional inequality
inequality in real analysis
mathematical inequality
result in mathematical analysis
Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Fenchel–Young inequality | 1 |
| Young inequality for products | 1 |
| Young's inequality canonical | 1 |
| Young's inequality for Orlicz spaces | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4092206 — 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: Young's inequality Context triple: [Jensen's inequality, relatedTo, Young's inequality]
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
- 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: Young's inequality Target entity description: Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
functional inequality
ⓘ
inequality in real analysis ⓘ mathematical inequality ⓘ result in mathematical analysis ⓘ |
| appliesTo |
Lebesgue integrable functions
ⓘ
measure spaces ⓘ nonnegative real numbers ⓘ |
| equalityHoldsIf |
a^p = b^q
ⓘ
a^p/p = b^q/q ⓘ |
| field |
convex analysis
ⓘ
functional analysis ⓘ mathematical analysis ⓘ real analysis ⓘ |
| hasApplication |
L^p space theory
ⓘ
Sobolev inequalities ⓘ
linked to:
Sobolev inequality
energy estimates in analysis ⓘ functional analysis of Banach spaces ⓘ information theory ⓘ interpolation inequalities ⓘ partial differential equations ⓘ probability theory ⓘ |
| hasCondition |
1/p + 1/q = 1
ⓘ
a \ge 0 ⓘ b \ge 0 ⓘ p > 1 ⓘ q > 1 ⓘ |
| hasForm | ab \le \frac{a^p}{p} + \frac{b^q}{q} for a,b \ge 0 and conjugate exponents p,q > 1 with 1/p + 1/q = 1 ⓘ |
| hasIntegralForm | \int fg \, d\mu \le \frac{1}{p}\int |f|^p d\mu + \frac{1}{q}\int |g|^q d\mu for conjugate exponents p,q ⓘ |
| hasProofMethod |
Jensen's inequality
ⓘ
linked to:
Jensen inequality
tangent line method for convex functions ⓘ |
| hasVariant |
Young's convolution inequality
ⓘ
linked to:
Young inequality for convolutions
Young's inequality for Orlicz spaces ⓘ
linked to:
Young's inequality
discrete form of Young's inequality ⓘ integral form of Young's inequality ⓘ |
| isBasedOn |
convexity of t \mapsto t^p
ⓘ
convexity of the exponential function ⓘ |
| isRelatedTo |
Hölder's inequality
ⓘ
linked to:
Hölder inequality
Legendre transform ⓘ
linked to:
Legendre transformation
Minkowski's inequality ⓘ
linked to:
Minkowski inequality
convex conjugate ⓘ |
| isSpecialCaseOf |
Fenchel–Young inequality
ⓘ
linked to:
Young's inequality
|
| isToolFor |
bounding products by sums of powers
ⓘ
deriving a priori estimates ⓘ establishing norm inequalities ⓘ estimating nonlinear terms in PDEs ⓘ |
| isUsedToProve |
Hölder's inequality
ⓘ
linked to:
Hölder inequality
Minkowski's inequality ⓘ
linked to:
Minkowski inequality
Young's convolution inequality ⓘ
linked to:
Young inequality for convolutions
|
| namedAfter | William Henry Young ⓘ |
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: Young's inequality Description of subject: Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Jensen inequality
linked to: Young's inequality
linked to: Young's inequality
linked to: Young's inequality