This article lists Wikipedia articles about named mathematical

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

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 *

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 \fr ...

* Babenko–Beckner inequality *

Bernoulli's inequality In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality (mathematics), inequality that approximates exponentiations of 1 + ''x''. It is often employed in real analysis. It has several useful variant ...

* Bernstein's inequality (mathematical analysis) *

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 The Bihari–LaSalle inequality, was proved by the American mathematician Joseph P. LaSalle (1916–1983) in 1949 and by the Hungarian mathematician Imre Bihari (1915–1998) in 1956. It is the following nonlinear generalization of Grönwall's lem ...

* Bohnenblust–Hille inequality *

Borell–Brascamp–Lieb inequality In mathematics, the Borell–Brascamp–Lieb inequality is an integral inequality due to many different mathematicians but named after Christer Borell, Herm Jan Brascamp and Elliott Lieb. The result was proved for ''p'' > 0 by Henst ...

* Brezis–Gallouet inequality * Carleman's inequality * 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 *

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 derivatives of the function and the geometry of the domain, a ...

* Gagliardo–Nirenberg interpolation inequality * Gårding's inequality *

Grothendieck inequality In mathematics, the Grothendieck inequality states that there is a universal constant K_G with the following property. If ''M'ij'' is an ''n'' × ''n'' (real or complex) matrix with : \Big, \sum_ M_ s_i t_j \Big, \le 1 for all (real ...

* Grunsky's inequalities *

Hanner's inequalities In mathematics, Hanner's inequalities are results in the theory of ''L'p'' spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of ''L'p'' spaces for ''p'' ∈ (1,&n ...

Hardy's inequality Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if a_1, a_2, a_3, \dots is a sequence of non-negative real numbers, then for every real number ''p'' > 1 one has :\sum_^\infty \left (\frac\right )^p\leq ...

Hardy–Littlewood inequality In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean sp ...

Hardy–Littlewood–Sobolev inequality In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...

Harnack's inequality In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functio ...

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 * Jensen's inequality * Khabibullin's conjecture on integral inequalities *

Kantorovich inequality In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality. The triangle inequality states that the length of two sides of any triangle, added to ...

* Karamata's inequality *

Korn's inequality In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric matrix, skew-symmetric at every point, then ...

* Ladyzhenskaya's inequality *

Landau–Kolmogorov inequality In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function ''f'' defined on a subset ''T'' of the real ...

* 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 ...

Minkowski's inequality In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the tr ...

Poincaré inequality In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry ...

Popoviciu's inequality In convex analysis, Popoviciu's inequality is an inequality about convex functions. It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu, a Romanian mathematician. Formulation Let ''f'' be a function from an interva ...

Prékopa–Leindler inequality In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is ...

Rayleigh–Faber–Krahn inequality In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eig ...

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 *

Schur test In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L^2\to L^2 operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X,\ ...

Shapiro inequality In mathematics, the Shapiro inequality is an inequality (mathematics), 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 ...

Sobolev inequality In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the ...

Steffensen's inequality Steffensen's inequality is an equation in mathematics named after Johan Frederik Steffensen. It is an integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other conce ...

* 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 In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors. If is ...

* Von Neumann's inequality * 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
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
s

* Hardy–Littlewood maximal inequality *

Inequality of arithmetic and geometric means In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...

* 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,&nbs ...

Mahler's inequality In mathematics, Mahler's inequality, named after Kurt Mahler, states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means: :\prod_^n (x ...

Muirhead's inequality In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. Preliminary definitions ''a''-mean For any real vector :a=(a_1,\dots ...

Newton's inequalities In mathematics, the Newton inequalities are named after Isaac Newton. Suppose ''a''1, ''a''2, ..., ''a'n'' are real numbers and let e_k denote the ''k''th elementary symmetric polynomial in ''a''1, ''a''2, ..., ''a ...

* 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 genet ...

* 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 leas ...

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 In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential inequality, differential or integral inequality by the soluti ...

Geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

{{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 Bonnesen's inequality is an inequality (mathematics), inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetry, isoperimetric ine ...

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 In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P ...

* 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 *

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 b ...

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 In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893.Maz'ya & Shaposhnikova It is a bound on the determinant of a matrix whose entries are complex numbe ...

* Hadwiger–Finsler inequality * Hinge theorem *

Hitchin–Thorpe inequality In differential geometry the Hitchin–Thorpe inequality is a relation which restricts the topology of 4-manifolds that carry an Einstein metric. Statement of the Hitchin–Thorpe inequality Let ''M'' be a closed, oriented, four-dimensio ...

Isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

* 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. Algo ...

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 ...

* Łojasiewicz inequality * Loomis–Whitney inequality * Melchior's inequality *

Milman's reverse Brunn–Minkowski inequality In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in '' ...

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 bun ...

Minkowski's first inequality for convex bodies In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality. State ...

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 Charle ...

* 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 ema ...

Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...

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 on ...

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 Inequalities are very important in the study of information theory. There are a number of different contexts in which these inequalities appear. Entropic inequalities Consider a tuple X_1,X_2,\dots,X_n of n finitely (or at most countably) supp ...

* Kraft's inequality *

Log sum inequality The log sum inequality is used for proving theorems in information theory. Statement Let a_1,\ldots,a_n and b_1,\ldots,b_n be nonnegative numbers. Denote the sum of all a_is by a and the sum of all b_is by b. The log sum inequality states that ...

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 ...

* Abhyankar's inequality * Pisier–Ringrose inequality

Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

* Abel's inequality *

Bregman–Minc inequality In discrete mathematics, the Bregman–Minc inequality, or Bregman's theorem, allows one to estimate the permanent of a binary matrix via its row or column sums. The inequality was conjectured in 1963 by Henryk Minc and first proved in 1973 by Le ...

* Cauchy–Schwarz inequality *

Golden–Thompson inequality In physics and mathematics, the Golden–Thompson inequality is a Trace inequalities, trace inequality between Matrix_exponential, exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the context o ...

* Hoffman-Wielandt inequality * Peetre's inequality * Sylvester's rank inequality *

* Trace inequalities

= Eigenvalue inequalities

= *

Bendixson's inequality In mathematics, Bendixson's inequality is a quantitative result in the field of Matrix_(mathematics), matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary and real parts of Eigenvalues_and_eigenvectors, characteris ...

* Weyl's inequality in matrix theory *

Cauchy interlacing theorem In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of Eigenvalues and eigenvectors, eigenvalues of ...

* Poincaré separation theorem

Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

Bonse's inequality In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if ''p''1, ..., ''p'n'', ''p'n''+1 are the s ...

* 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) In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let ''X''1, ..., ''X'n'' be independent Bernoulli random variables taking value ...

Boole's inequality In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individu ...

* Borell–TIS inequality * BRS-inequality * Burkholder's inequality * Burkholder–Davis–Gundy inequalities * Cantelli's inequality *

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 th ...

* Chernoff's inequality * Chung–Erdős inequality *

Concentration inequality In probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value). The law of large numbers of classical probability theory states that sums of independent random vari ...

* Cramér–Rao inequality *

Doob's martingale inequality In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale exceeds any given value over a given inter ...

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 In information theory, the entropy power inequality (EPI) is a result that relates to so-called "entropy power" of random variables. It shows that the entropy power of suitably well-behaved random variables is a superadditive function. The entrop ...

Etemadi's inequality In probability theory, Etemadi's inequality is a so-called "maximal inequality", an inequality that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. The result ...

* Fannes–Audenaert inequality * Fano's inequality * Fefferman's inequality * Fréchet inequalities *

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'', a ...

Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the ...

, the statement that the least-squares estimators in certain linear models are the best linear unbiased estimators *

Gaussian correlation inequality The Gaussian correlation inequality (GCI), formerly known as the Gaussian correlation conjecture (GCC), is a mathematical theorem in the fields of mathematical statistics and convex geometry. The statement The Gaussian correlation inequality stat ...

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, half-s ...

* Gibbs's inequality *

Hoeffding's inequality In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Was ...

* Hoeffding's lemma * 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 In probability theory, Kolmogorov's inequality is a so-called "maximal inequality (mathematics), inequality" that gives a bound on the probability that the partial sums of a Finite set, finite collection of independent random variables exceed some s ...

Kunita–Watanabe inequality In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the Cauchy–Schwarz inequality to integrals of stochastic processes. It was first obtained by Hiroshi Kunita and Shinzo Watanabe and plays a fundamental role in thei ...

* Le Cam's theorem * Lenglart's inequality * Marcinkiewicz–Zygmund inequality *

Markov's inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, ...

* McDiarmid's inequality *

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 moment (mathematics), moments. The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If ''Z' ...

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 tig ...

* 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 In queueing theory, a discipline within the mathematical theory of probability, Ross's conjecture gives a lower bound for the average waiting-time experienced by a customer when arrivals to the queue do not follow the simplest model for random arriv ...

, a lower bound on the average waiting time in certain queues *

Samuelson's inequality In statistics, Samuelson's inequality, named after the economist Paul Samuelson, also called the Laguerre–Samuelson inequality, after the mathematician Edmond Laguerre, states that every one of any collection ''x''1, ..., ''x'n'' ...

Shearer's inequality Shearer's inequality or also Shearer's lemma, in mathematics, is an inequality in information theory relating the Entropy (information theory), entropy of a set of variables to the entropies of a collection of subsets. It is named for mathematician ...

* Stochastic Gronwall inequality *

Talagrand's concentration inequality In the probability theory field of mathematics , Talagrand's concentration inequality is an isoperimetric-type inequality (mathematics), inequality for product space, product probability spaces. It was first proved by the French mathematician Miche ...

Vitale's random Brunn–Minkowski inequality In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of ''n''-dimensional Euclidean space R''n'' to random compact sets. ...

* Vysochanskiï–Petunin inequality

Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

* Berger's inequality for Einstein manifolds

Inequalities particular to physics

Ahlswede–Daykin inequality The Ahlswede–Daykin inequality , also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilis ...

* 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 The Clausius–Duhem inequality is a way of expressing the second law of thermodynamics that is used in continuum mechanics. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamicall ...

* Correlation inequality – any of several inequalities * FKG inequality * Ginibre inequality * Griffiths inequality *

Heisenberg's inequality In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values fo ...

* Holley inequality * Leggett–Garg inequality * Riemannian Penrose inequality * Rushbrooke inequality *

Tsirelson's inequality A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates Bell inequalities (i.e., it cannot be described by a local hidden-variable theory), a natural question to ask is h ...