Analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

• Agmon's inequality
  Agmon's inequality
  In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of a few closely related inequalities between the Lebesgue space L^\infty and the Sobolev spaces H^s. It is useful in the study of partial differential equations....
  
  
• Askey–Gasper 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:...
  
  
• Babenko–Beckner inequality
  Babenko–Beckner inequality
  In mathematics, the Babenko–Beckner inequality is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces...
  
  
• Bernoulli's inequality
  Bernoulli's inequality
  In real analysis, Bernoulli's inequality is an inequality that approximates exponentiations of 1 + x.The inequality states that^r \geq 1 + rx\!...
  
  
• Bernstein's inequality (mathematical analysis)
• Bessel's inequality
  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....
  
  
• Bihari's inequality
  Bihari's inequality
  Bihari's inequality, proved by Hungarian mathematician Imre Bihari , is the following nonlinear generalization of the Grönwall's lemma....
  
  
• Borell–Brascamp–Lieb inequality
• Carleman's 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 Carleman's condition for the determinancy of the problem of moments.-Statement:Let a1, a2, a3, .....
  
  
• Chebyshev–Markov–Stieltjes inequalities
• Chebyshev's sum inequality
  Chebyshev's sum inequality
  In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that ifa_1 \geq a_2 \geq \cdots \geq a_nandb_1 \geq b_2 \geq \cdots \geq b_n,then...
  
  
• Clarkson's inequalities
  Clarkson's inequalities
  In mathematics, Clarkson's inequalities are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions in Lp in terms of the Lp-norms of those functions individually.-Statement of the inequalities:Let be a measure space; let...
  
  
• Eilenberg's inequality
  Eilenberg's inequality
  Eilenberg's inequality is a mathematical inequality for Lipschitz-continuous functions.Let ƒ : X → Y be a Lipschitz-continuous function between separable metric spaces whose Lipschitz constant is denoted by Lip ƒ...
  
  
• Fekete–Szegő inequality
  Fekete–Szegő inequality
  In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by , related to the Bieberbach conjecture...
  
  
• Fenchel's inequality
• Friedrichs' inequality
  Friedrichs' inequality
  In mathematics, Friedrichs' 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, and can be used to show that certain norms on Sobolev spaces are...
  
  
• Gårding's inequality
  Gårding's inequality
  In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator...
  
  
• Grothendieck inequality
• Hanner's inequalities
  Hanner's inequalities
  In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces for p ∈  than the approach proposed by James Clarkson in 1936.-Statement of the...
  
  
• Hardy's inequality
• Harnack's inequality
  Harnack's inequality
  In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . and generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Perelman's solution of the Poincaré conjecture uses...
  
  
• Hausdorff–Young inequality
  Hausdorff–Young inequality
  In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. proved the inequality for some special values of q, and proved it in general...
  
  
• Hermite–Hadamard inequality
  Hermite–Hadamard inequality
  In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : [a, b] → R is convex, then...
  
  
• Hilbert's inequality
  Hilbert's inequality
  In complex analysis, a branch of mathematics, Hilbert's inequality was first demonstrated by David Hilbert and later improved by Issai Schur, who replaced Hilbert's constant 2\pi with the sharp constant \pi....
  
  
• Hölder's inequality
  Hölder's inequality
  In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
  
  
• Jackson'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 of its derivatives...
  
  
• Jensen's inequality
  Jensen's inequality
  In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context,...
  
  
• Khabibullin’s conjecture on integral inequalities
  Khabibullin’s conjecture on integral inequalities
  In mathematics, Khabibullin's conjecture, named after B. N. Khabibullin, is related to Paley's problem for plurisubharmonic functions and to various extremal problems in the theory of entire functions of several variables....
  
  
• Kantorovich inequality
  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....
  
  
• Korn's inequality
  Korn's inequality
  In mathematics, Korn's inequality is a result about the derivatives of Sobolev functions. Korn's inequality plays an important role in linear elasticity theory.-Statement of the inequality:...
  
  
• Landau–Kolmogorov inequality
• Lebedev–Milin inequality
  Lebedev–Milin inequality
  In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by and...
  
  
• Markov brothers' inequality
  Markov brothers' inequality
  In mathematics, the Markov brothers' inequality is an inequality proved by Andrey Markov and Vladimir Markov. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. For k = 1 it was proved by Andrey Markov, and for k = 2,3,.....
  
  
• Max-min inequality
• Minkowski's inequality
• Poincaré inequality
  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 of its domain of definition. Such bounds are of great...
  
  
• Popoviciu's inequality
  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. It states:...
  
  
• Prékopa–Leindler inequality
• Rayleigh–Faber–Krahn inequality
• Remez inequality
  Remez inequality
  In mathematics, the Remez inequality, discovered by the Ukrainian mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.-The inequality:...
  
  
• Shapiro inequality
  Shapiro inequality
  In mathematics, the Shapiro inequality is an inequality proposed by H. 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, or...
  
  
• Sobolev inequality
  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 Rellich–Kondrachov theorem showing that under...
  
  
• Steffensen's inequality
• Szegő inequality
  Szegő inequality
  In functional analysis, a mathematical discipline, the Szegő inequality or Pólya–Szegő inequality, named after George Pólya and Gábor Szegő, states that if...
  
  
• Trudinger's theorem
  Trudinger's theorem
  In mathematical analysis, Trudinger's theorem or the Trudinger inequality is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger ....
  
  
• Turán's inequalities
  Turán's inequalities
  In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by . There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors....
  
  
• Von Neumann's inequality
  Von Neumann's inequality
  In operator theory, von Neumann's inequality, due to John von Neumann, states that, for a contraction T acting on a Hilbert space and a polynomial p, then the norm of p is bounded by the supremum of |p| for z in the unit disk." In other words, for a fixed contraction T, the polynomial functional...
  
  
• Wirtinger's inequality for functions

Inequalities relating to mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

s

• Hardy–Littlewood maximal inequality
• Inequality of arithmetic and geometric means
  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 further, that the two means are equal if and only if...
  
  
• Ky Fan inequality
  Ky Fan inequality
  In mathematics, there are two different results that share the common name of the Ky Fan inequality. One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book Inequalities by...
  
  
• Levinson's inequality
• Maclaurin's inequality
• Mahler's inequality
  Mahler's inequality
  In mathematics, Mahler's inequality 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:...
  
  
• Muirhead's inequality
  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.-The "a-mean":For any real vectora=...
  
  
• Newton's inequalities
  Newton's inequalities
  In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are real numbers and let \sigma_k denote the kth elementary symmetric function in a1, a2, ..., an...
  
  
• Stein–Strömberg theorem
• Young's inequality
  Young's inequality
  In mathematics, the term Young's inequality is used for two inequalities: one about the product of two numbers, and one about the convolution of two functions. They are named for William Henry Young....
  
  

Combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

• Fishburn–Shepp inequality
  Fishburn–Shepp inequality
  In combinatorial mathematics, the Fishburn–Shepp inequality is an inequality for the number of extensions of partial orders to linear orders, found by and .It states that if x, y, and z are incomparable elements of a finite poset, then PIn combinatorial mathematics, the Fishburn–Shepp inequality...
  
  
• Fisher's inequality
  Fisher's inequality
  In combinatorial mathematics, Fisher's inequality, named after Ronald Fisher, is a necessary condition for the existence of a balanced incomplete block design satisfying certain prescribed conditions....
  
  
• Lubell–Yamamoto–Meshalkin inequality
• Nesbitt's inequality
• Rearrangement inequality
• Schur's inequality
  Schur's inequality
  In mathematics, Schur's inequality, named after Issai Schur,establishes that for all non-negative real numbersx, y, z and a positive number t,x^t + y^t + z^t \ge 0...
  
  

Geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

• Alexandrov–Fenchel inequality
• Aristarchus' inequality
  Aristarchus' inequality
  In trigonometry, Aristarchus' inequality, named after the ancient astronomer Aristarchus of Samos, states that if α and β are acute angles and β ...
  
  
• Barrow's inequality
  Barrow's inequality
  In geometry, Barrow's inequality states the following: Let P be a point inside the triangle ABC; U, V, and W be the points where the angle bisectors of BPC, CPA, and APB intersect the sides BC, CA, AB, respectively...
  
  
• Berger–Kazdan comparison theorem
• Blaschke–Santaló inequality
• Bishop–Gromov 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 theorem....
  
  
• Bogomolov–Miyaoka–Yau inequality
  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 underlying real 4-manifold...
  
  
• Bonnesen's inequality
  Bonnesen's inequality
  Bonnesen's inequality is an 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 isoperimetric inequality....
  
  
• Brascamp–Lieb inequality
• Brunn–Minkowski inequality
• Cheng's eigenvalue comparison theorem
  Cheng's eigenvalue comparison theorem
  In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also...
  
  
• Clifford's theorem on special divisors
• Erdős–Mordell inequality
  Erdos–Mordell inequality
  In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point O inside ABC, the sum of the distances from O to the sides is less than or equal to half of the sum of the distances from O to the vertices. It is named after Paul Erdős and Louis Mordell. posed the...
  
  
• Euler's theorem in geometry
  Euler's theorem in geometry
  In geometry, Euler's theorem, named after Leonhard Euler, states that the distance d between the circumcentre and incentre of a triangle can be expressed as d^2=R \,...
  
  
• Gromov's inequality for complex projective space
• Gromov's systolic inequality for essential manifolds
  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...
  
  
• Hadamard's inequality
  Hadamard's inequality
  In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors...
  
  
• Hadwiger–Finsler inequality
  Hadwiger–Finsler inequality
  In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler...
  
  
• Hinge theorem
  Hinge theorem
  In geometry, the hinge theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second...
  
  
• Hitchin–Thorpe inequality
  Hitchin–Thorpe inequality
  In differential geometry the Hitchin–Thorpe inequality is a famous relation which restricts the topology of 4-manifolds that carry an Einstein metric.- Statement of the Hitchin–Thorpe inequality :...
  
  
• Isoperimetric inequality
  Isoperimetry
  The isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. Isoperimetric literally means "having the same perimeter"...
  
  
• Jordan's inequality
  Jordan's inequality
  In mathematics, Jordan's inequality, named after Camille Jordan, states thatIt can be proven through the geometry of circles....
  
  
• Jung's theorem
  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...
  
  
• Loewner's torus inequality
  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:...
  
  
• Łojasiewicz inequality
  Łojasiewicz inequality
  In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real-analytic function on an open set U in Rn,...
  
  
• Loomis–Whitney inequality
• Melchior's inequality
• Milman's reverse Brunn–Minkowski inequality
• Minkowski's first inequality for convex bodies
  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....
  
  
• Myers's theorem
• Noether inequality
  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...
  
  
• Ono's inequality
  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, as shown by Balitrand in 1916....
  
  
• Pedoe's inequality
  Pedoe's inequality
  In geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then...
  
  
• Ptolemy's inequality
• Pu's inequality
• Riemannian Penrose inequality
  Riemannian Penrose inequality
  In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special...
  
  
• Toponogov's theorem
  Toponogov's theorem
  In the mathematical field of Riemannian geometry, Toponogov's theorem is a triangle comparison theorem. It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature than they would in a region...
  
  
• Triangle inequality
  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 ....
  
  
• Weitzenböck's inequality
  Weitzenböck's inequality
  In mathematics, Weitzenböck's inequality states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds:Equality occurs if and only if the triangle is equilateral...
  
  
• Wirtinger inequality (2-forms)
  Wirtinger inequality (2-forms)
  In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that the exterior \scriptstyle\nuth power of the standard symplectic form ω, when evaluated on a simple -vector ζ of unit volume, is bounded above by \scriptstyle\nu!...
  
  

Information theory

Information theory

Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

• Inequalities in 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.-Shannon-type inequalities:...
  
  
• Kraft's inequality
  Kraft's inequality
  In coding theory, Kraft's inequality, named after Leon Kraft, gives a necessary and sufficient condition for the existence of a uniquely decodable code for a given set of codeword lengths...
  
  
• Log sum inequality
  Log sum inequality
  In mathematics, the log sum inequality is an inequality which is useful for proving several 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_i\;s by a and the sum of all b_i\;s by b...
  
  
• Welch bounds
  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 engineering, particularly in coding theory. The bounds...
  
  

Linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

• Abel's inequality
  Abel's inequality
  In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case....
  
  
• Cauchy–Schwarz inequality
  Cauchy–Schwarz inequality
  In mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...
  
  
• Golden–Thompson inequality
• Peetre's inequality
  Peetre's inequality
  In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number t and any vectors x and y in Rn, the following inequality holds: \left^t \le 2^...
  
  
• Triangle inequality
  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 ....
  
  
• von Neumann's trace inequality
• Weyl's inequality

Number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

• Bonse's inequality
  Bonse's inequality
  In number theory, Bonse's inequality, named after H. Bonse, states that if p1, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then...
  
  
• Large sieve inequality
• Pólya–Vinogradov inequality
• Turán–Kubilius inequality
  Turán–Kubilius inequality
  The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function. The theorem was proved in a special case in 1934 by Paul Turán and generalized in 1956 and 1964 by Jonas Kubilius.-Statement of...
  
  
• Weyl's inequality

Probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 

• Azuma's inequality
• Bennett's inequality
  Bennett's inequality
  In probability theory, Bennett's inequality provides 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
  Bhatia–Davis inequality
  In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance of any bounded probability distribution on the real line....
  
  
• Bernstein inequalities (probability theory)
• Boole's inequality
  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 individual events...
  
  
• Chebyshev's inequality
  Chebyshev's inequality
  In probability theory, Chebyshev’s inequality guarantees that in any data sample or probability distribution,"nearly all" values are close to the mean — the precise statement being that no more than 1/k2 of the distribution’s values can be more than k standard deviations away from the mean...
  
  
• Chernoff's inequality
• Concentration inequality
  Concentration inequality
  In mathematics, concentration inequalities provide probability bounds on how a random variable deviates from some value . The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a...
  
  
• Cramér–Rao inequality
• Doob's martingale inequality
  Doob's martingale inequality
  In mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time...
  
  
• Dvoretzky–Kiefer–Wolfowitz inequality
  Dvoretzky–Kiefer–Wolfowitz inequality
  In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz inequality predicts how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn...
  
  
• Gauss–Markov theorem
  Gauss–Markov theorem
  In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator of the coefficients is given by the...
  
  
• Entropy power inequality
  Entropy power inequality
  In mathematics, the entropy power inequality is a result in probability theory 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 entropy power inequality was proved in 1948 by...
  
  
• Etemadi's inequality
  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...
  
  
• Fannes–Audenaert inequality
  Fannes–Audenaert inequality
  The Fannes–Audenaert inequality is a mathematical bound on the difference between the von Neumann entropies of two density matrices as a function of their trace distance. It was proved by Koenraad M. R. Audenaert in 2007 as an optimal refinement of M...
  
  
• Fano's inequality
  Fano's inequality
  In information theory, Fano's inequality relates the average information lost in a noisy channel to the probability of the categorization error. It was derived by Robert Fano in the early 1950s while teaching a Ph.D...
  
  
• Gauss's inequality
  Gauss's inequality
  In probability theory, Gauss's inequality gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode....
  
  
• Gaussian isoperimetric inequality
• Gibbs' inequality
  Gibbs' inequality
  In information theory, Gibbs' inequality is a statement about the mathematical entropy of a discrete probability distribution. Several other bounds on the entropy of probability distributions are derived from Gibbs' inequality, including Fano's inequality....
  
  
• Hoeffding's inequality
  Hoeffding's inequality
  In probability theory, Hoeffding's inequality provides an upper bound on the probability for the sum of random variables to deviate from its expected value. Hoeffding's inequality was proved by Wassily Hoeffding.LetX_1, \dots, X_n \!...
  
  
• Hoeffding's lemma
  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....
  
  
• Jensen's inequality
  Jensen's inequality
  In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context,...
  
  
• Khintchine inequality
  Khintchine inequality
  In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Roman alphabet, is a theorem from probability, and is also frequently used in analysis...
  
  
• Kolmogorov's inequality
  Kolmogorov's inequality
  In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound...
  
  
• Le Cam's theorem
  Le Cam's theorem
  In probability theory, Le Cam's theorem, named after Lucien le Cam , is as follows.Suppose:* X1, ..., Xn are independent random variables, each with a Bernoulli distribution , not necessarily identically distributed.* Pr = pi for i = 1, 2, 3, ...* \lambda_n = p_1 + \cdots + p_n.\,* S_n = X_1...
  
  
• Marcinkiewicz–Zygmund inequality
  Marcinkiewicz–Zygmund inequality
  In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables...
  
  
• Markov's 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...
  
  
• McDiarmid's inequality
• Paley–Zygmund inequality
  Paley–Zygmund inequality
  In mathematics, the Paley–Zygmund inequality bounds theprobability that a positive random variable is small, in terms ofits mean and variance...
  
  
• Pinsker's inequality
  Pinsker's inequality
  In information theory, Pinsker's inequality, named after its inventor Mark Semenovich Pinsker, is an inequality that relates Kullback-Leibler divergence and the total variation distance...
  
  
• Popoviciu's inequality on variances
  Popoviciu's inequality on variances
  In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution...
  
  
• Rao–Blackwell theorem
  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-squared-error criterion or any of a variety of similar...
  
  
• Samuelson's inequality
  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 x1, ..., xn, is within √ standard deviations of their mean...
  
  
• Shearer's inequality
• Simpson's paradox
  Simpson's paradox
  In probability and statistics, Simpson's paradox is a paradox in which a correlation present in different groups is reversed when the groups are combined. This result is often encountered in social-science and medical-science statistics, and it occurs when frequencydata are hastily given causal...
  
  
• Talagrand's concentration inequality
  Talagrand's concentration inequality
  In probability theory, Talagrand's concentration inequality, is an isoperimetric-type inequality for product probability spaces. It was first proved by the French mathematician Michel Talagrand...
  
  
• Vitale's random Brunn–Minkowski inequality
• Vysochanskiï–Petunin inequality

Inequalities particular to physics

• Ahlswede–Daykin inequality
  Ahlswede–Daykin inequality
  A fundamental tool in statistical mechanics and probabilistic combinatorics , the Ahlswede–Daykin inequality is a correlation-type inequality for four functions on a finite distributive lattice....
  
  
• Bell's inequality – see Bell's theorem
  Bell's theorem
  In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
  
  
  • Bell's original inequality
• CHSH inequality
  CHSH inequality
  In physics, the CHSH Bell test is an application of Bell's theorem, intended to distinguish between the entanglement hypothesis of quantum mechanics and local hidden variable theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony and Richard Holt, who described it in a much-cited...
  
  
• Clausius–Duhem inequality
• Correlation inequality
  Correlation inequality
  In probability and statistics, a correlation inequality is one of a number of inequalities satisfied by the correlation functions of a model. Such inequalities are of particular use in statistical mechanics and in percolation theory.Examples include:...
  
   – any of several inequalities
• FKG inequality
  FKG inequality
  The FKG inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics , due to...
  
  
• Ginibre inequality
• Griffiths inequality
  Griffiths inequality
  In statistical mechanics, the Griffiths inequality , named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems...
  
  
• Heisenberg's inequality
• Holley inequality
• Leggett–Garg inequality
• Riemannian Penrose inequality
  Riemannian Penrose inequality
  In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special...
  
  
• Rushbrooke inequality
  Rushbrooke inequality
  In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature T....
  
  
• Sakurai's Bell inequality
  Sakurai's Bell inequality
  The intention of a Bell inequality is to serve as a test of local realism or local hidden variable theories as against quantum mechanics, applying Bell's theorem, which shows them to be incompatible. Not all the Bell's inequalities that appear in the literature are in fact fit for this purpose...
  
  
• Tsirelson's inequality
• Wigner–d'Espagnat inequality