This article lists Wikipedia articles about named mathematical inequalities.

Inequalities in pure mathematics

Analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

Agmon's inequality In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,Lemma 13.2, in: Agmon, Shmuel, ''Lectures on Elliptic Boundary Value Problems'', AMS Chelsea Publishing, Providence, RI, 2010. . consist of two closely related interpolatio ...

Askey–Gasper inequality In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture. Statement It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then :\sum_^n \ ...

* Babenko–Beckner inequality *

Bernoulli's inequality In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + ''x''. It is often employed in real analysis. It has several useful variants: * (1 + x)^r \geq 1 + r ...

Bernstein's inequality (mathematical analysis) Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation ...

Bessel's inequality In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let H be a Hi ...

* Bihari–LaSalle inequality * Bohnenblust–Hille inequality * Borell–Brascamp–Lieb inequality * Brezis–Gallouet inequality *

Carleman's inequality Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923 and used it to prove the Denjoy–Carleman theorem on quasi-analytic classes. Statement Let a_1,a_2,a_3,\dots be a sequence of non-ne ...

* Chebyshev–Markov–Stieltjes inequalities *

Chebyshev's sum inequality In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if :a_1 \geq a_2 \geq \cdots \geq a_n \quad and \quad b_1 \geq b_2 \geq \cdots \geq b_n, then : \sum_^n a_k b_k \geq \left(\sum_^n a_k\right)\!\!\left(\sum_^n ...

* Clarkson's inequalities * Eilenberg's inequality * Fekete–Szegő inequality *

Fenchel's inequality In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformatio ...

Friedrichs's inequality In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the ''Lp'' norm of a function using ''Lp'' bounds on the weak derivative In mathematics, a weak derivative is a general ...

* Gagliardo–Nirenberg interpolation inequality * Gårding's inequality * Grothendieck inequality * Grunsky's inequalities * Hanner's inequalities * Hardy's inequality * Hardy–Littlewood inequality * Hardy–Littlewood–Sobolev inequality * Harnack's inequality *

Hausdorff–Young inequality The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by . It is now typically understood as a rather direct corollary of th ...

* Hermite–Hadamard inequality * Hilbert's inequality * Hölder's inequality *

Jackson's inequality In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of its ...

* Jensen's inequality * Khabibullin's conjecture on integral inequalities * Kantorovich inequality * Karamata's inequality * Korn's inequality * Ladyzhenskaya's inequality * Landau–Kolmogorov inequality * Lebedev–Milin inequality * Lieb–Thirring inequality * Littlewood's 4/3 inequality * Markov brothers' inequality * Mashreghi–Ransford inequality *

Max–min inequality In mathematics, the max–min inequality is as follows: :For any function \ f : Z \times W \to \mathbb\ , :: \sup_ \inf_ f(z, w) \leq \inf_ \sup_ f(z, w)\ . When equality holds one says that , , and satisfies a strong max–min property (or a s ...

* Minkowski's inequality * Poincaré inequality * Popoviciu's inequality * Prékopa–Leindler inequality * Rayleigh–Faber–Krahn inequality *

Remez inequality In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials. The inequality Let ''σ'' be an ...

Riesz rearrangement inequality In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f : \mathbb^n \to \mathbb^+, g : \mathbb^n \to \mathbb^+ and h : \mathbb^n \to \mathbb^+ satisfy the inequ ...

* Schur test *

Shapiro inequality In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954. Statement of the inequality Suppose n is a natural number and x_1, x_2, \dots, x_n are positive numbers and: * n is even and less than or equal to 12 ...

* Sobolev inequality * Steffensen's inequality * Szegő inequality * Three spheres inequality * Trace inequalities *

Trudinger's theorem In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provid ...

* Turán's inequalities *

Von Neumann's inequality In operator theory, von Neumann's inequality, due to John von Neumann, states that, for a fixed contraction ''T'', the polynomial functional calculus map is itself a contraction. Formal statement For a contraction ''T'' acting on a Hilbert spac ...

* Wirtinger's inequality for functions *

Young's convolution inequality In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young. Statement Euclidean Space In real analysis, the following result is called Young's convolution ...

Young's inequality for products In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality f ...

Inequalities relating to means

* Hardy–Littlewood maximal inequality * Inequality of arithmetic and geometric means * Ky Fan inequality * Levinson's inequality *

Maclaurin's inequality In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let ''a''1, ''a''2, ..., ''a'n'' be positive real numbers, and for ''k'' = 1,&n ...

* Mahler's inequality * Muirhead's inequality * Newton's inequalities * Stein–Strömberg theorem

Combinatorics

* Binomial coefficient bounds * Factorial bounds * XYZ inequality *

Fisher's inequality Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population gen ...

* Ingleton's inequality * Lubell–Yamamoto–Meshalkin inequality *

Nesbitt's inequality In mathematics, Nesbitt's inequality states that for positive real numbers ''a'', ''b'' and ''c'', :\frac+\frac+\frac\geq\frac. It is an elementary special case (N = 3) of the difficult and much studied Shapiro inequality, and was published at l ...

Rearrangement inequality In mathematics, the rearrangement inequality states that x_n y_1 + \cdots + x_1 y_n \leq x_ y_1 + \cdots + x_ y_n \leq x_1 y_1 + \cdots + x_n y_n for every choice of real numbers x_1 \leq \cdots \leq x_n \quad \text \quad y_1 \leq \cdots \leq y_n ...

* Schur's inequality *

* Stirling's formula (bounds)

Differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s

* Grönwall's inequality

Geometry

{{See also, List of triangle inequalities * Alexandrov–Fenchel inequality *

Aristarchus's inequality Aristarchus's inequality (after the Greek astronomer and mathematician Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of trigonometry which states that if ''α'' and ''β'' are acute angles (i.e. between 0 and a right angle) an ...

* Barrow's inequality * Berger–Kazdan comparison theorem * Blaschke–Lebesgue inequality * Blaschke–Santaló inequality *

Bishop–Gromov inequality In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness ...

Bogomolov–Miyaoka–Yau inequality In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality : c_1^2 \le 3 c_2 between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the under ...

* Bonnesen's inequality *

Brascamp–Lieb inequality In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on ''n''-dimensional Euclidean space \mathbb^. It generalizes the Loomis–Whitney inequality and Höl ...

* Brunn–Minkowski inequality * Castelnuovo–Severi inequality * Cheng's eigenvalue comparison theorem * Clifford's theorem on special divisors * Cohn-Vossen's inequality *

Erdős–Mordell inequality In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ''ABC'' and point ''P'' inside ''ABC'', the sum of the distances from ''P'' to the sides is less than or equal to half of the sum of the distances from ''P'' to the ...

Euler's theorem in geometry In geometry, Euler's theorem states that the distance ''d'' between the circumcenter and incenter of a triangle is given by d^2=R (R-2r) or equivalently \frac + \frac = \frac, where R and r denote the circumradius and inradius respectively (the ...

Gromov's inequality for complex projective space In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality : \mathrm_2^n \leq n! \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by ...

Gromov's systolic inequality for essential manifolds In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1 ...

* Hadamard's inequality *

Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ + ...

* Hinge theorem * Hitchin–Thorpe inequality * Isoperimetric inequality * Jordan's inequality *

Jung's theorem In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Alg ...

Loewner's torus inequality In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus. Statement In 1949 Charles Loewner proved that every metric on th ...

* Łojasiewicz inequality *

Loomis–Whitney inequality In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d-dimensional set by the sizes of its (d-1)-dimensional projections. The inequality has applications in inci ...

* Melchior's inequality * Milman's reverse Brunn–Minkowski inequality *

Milnor–Wood inequality In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after John Milnor and John W. Wood. Flat b ...

* Minkowski's first inequality for convex bodies *

Myers's theorem Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following: In the special case of ...

Noether inequality In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfac ...

Ono's inequality In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, ...

Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a triangle with area '' ...

Ptolemy's inequality In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space. It states that, for any four points , , , and , the following inequality holds: :\overline\cdot \overli ...

Pu's inequality In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it. Statement A student of Charl ...

* Riemannian Penrose inequality *

Toponogov's theorem In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem. It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics em ...

* Triangle inequality *

Weitzenböck's inequality In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt\, \Delta. Equality occurs if and o ...

Wirtinger inequality (2-forms) : ''For other inequalities named after Wirtinger, see Wirtinger's inequality.'' In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold , the exterior th power of the symplectic form ...

Information theory

* Inequalities in information theory * Kraft's inequality * Log sum inequality *

Welch bounds In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engin ...

Algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

* Abhyankar's inequality * Pisier–Ringrose inequality

Linear algebra

* Abel's inequality * Bregman–Minc inequality *

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...

* Golden–Thompson inequality * Hadamard's inequality * Hoffman-Wielandt inequality * Peetre's inequality * Sylvester's rank inequality * Triangle inequality * Trace inequalities

= Eigenvalue inequalities

= * Bendixson's inequality * Weyl's inequality in matrix theory * Cauchy interlacing theorem * Poincaré separation theorem

Number theory

* Bonse's inequality * Large sieve inequality * Pólya–Vinogradov inequality * Turán–Kubilius inequality *

Weyl's inequality In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix. Weyl's inequality about perturbation Let ...

Probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and statistics

Azuma's inequality In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences. Suppose \ is a martingale (or super-martingale) ...

* Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount * Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution * Bernstein inequalities (probability theory) * Boole's inequality * Borell–TIS inequality * BRS-inequality * Burkholder's inequality * Burkholder–Davis–Gundy inequalities *

Cantelli's inequality In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for \lambda > ...

Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...

Chernoff's inequality In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Despite being named after Herman Chernoff, the author of the paper it first appeared in, the result is d ...

* Chung–Erdős inequality * Concentration inequality * Cramér–Rao inequality * Doob's martingale inequality *

Dvoretzky–Kiefer–Wolfowitz inequality In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz–Massart inequality (DKW inequality) bounds how close an empirically determined distribution function will be to the distribution function from which the empirical ...

* Eaton's inequality, a bound on the largest absolute value of a linear combination of bounded random variables * Emery's inequality * Entropy power inequality * Etemadi's inequality * Fannes–Audenaert inequality * Fano's inequality * Fefferman's inequality *

Fréchet inequalities In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George BooleBoole, G. (1854). ''An Investigation of the Laws of Thought, On Which Are Founded the Mathematical Theo ...

Gauss's inequality In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode. Let ''X'' be a unimodal random variable with mode ''m'', ...

* Gauss–Markov theorem, the statement that the least-squares estimators in certain linear models are the best linear unbiased estimators * Gaussian correlation inequality *

Gaussian isoperimetric inequality In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the ''n''-dimensional Euclidean space, hal ...

* Gibbs's inequality * Hoeffding's inequality *

Hoeffding's lemma In probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable. It is named after the Finnish–American mathematical statistician Wassily Hoeffding. The proof of Hoeff ...

* Jensen's inequality *

Khintchine inequality In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N complex ...

* Kolmogorov's inequality * Kunita–Watanabe inequality *

Le Cam's theorem In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following. Suppose: * X_1, X_2, X_3, \ldots are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), no ...

* Lenglart's inequality * Marcinkiewicz–Zygmund inequality * Markov's inequality *

McDiarmid's inequality In probability theory and theoretical computer science, McDiarmid's inequality is a concentration inequality which bounds the deviation between the sampled value and the expected value of certain functions when they are evaluated on independent ra ...

Paley–Zygmund inequality In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If ''Z'' ≥ 0 is ...

Pinsker's inequality In information theory, Pinsker's inequality, named after its inventor Mark Semenovich Pinsker, is an inequality that bounds the total variation distance (or statistical distance) in terms of the Kullback–Leibler divergence. The inequality is tigh ...

* Popoviciu's inequality on variances * Prophet inequality *

Rao–Blackwell theorem In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squ ...

* Ross's conjecture, a lower bound on the average waiting time in certain queues * Samuelson's inequality * Shearer's inequality * Stochastic Gronwall inequality * Talagrand's concentration inequality * Vitale's random Brunn–Minkowski inequality * Vysochanskiï–Petunin inequality

Topology

* Berger's inequality for Einstein manifolds

Inequalities particular to physics

* Ahlswede–Daykin inequality * Bell's inequality – see Bell's theorem ** Bell's original inequality *

CHSH inequality In physics, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics can not be reproduced by local hidden-variable theories. Experimental verification of the i ...

* Clausius–Duhem inequality * Correlation inequality – any of several inequalities * FKG inequality * Ginibre inequality * Griffiths inequality * Heisenberg's inequality * Holley inequality *

Leggett–Garg inequality The Leggett–Garg inequality, named for Anthony James Leggett and Anupam Garg, is a mathematical inequality fulfilled by all macrorealistic physical theories. Here, macrorealism (macroscopic realism) is a classical worldview defined by the conjunc ...

* Riemannian Penrose inequality * Rushbrooke inequality * Tsirelson's inequality