List of inequalities

Analysis

Agmon's inequality

Askey–Gasper inequality

Babenko–Beckner inequality

Bernoulli's inequality

Bernstein's inequality (mathematical analysis)

Bessel's inequality

Bihari–LaSalle inequality

Bohnenblust-Hille inequality

Borell–Brascamp–Lieb inequality

Brezis–Gallouet inequality

Carleman's inequality

Chebyshev–Markov–Stieltjes inequalities

Chebyshev's sum inequality

Clarkson's inequalities

Eilenberg's inequality

Fekete–Szegő inequality

Fenchel's inequality

Friedrichs' inequality

Gagliardo–Nirenberg interpolation inequality

Gårding's inequality

Grothendieck inequality

Grunsky's inequalities

Hanner's inequalities

Hardy's inequality

Hardy–Littlewood inequality

Harnack's inequality

Hausdorff–Young inequality

Hermite–Hadamard inequality

Hilbert's inequality

Hölder's inequality

Jackson's inequality

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

Minkowski's inequality

Poincaré inequality

Popoviciu's inequality

Prékopa–Leindler inequality

Rayleigh–Faber–Krahn inequality

Remez inequality

Riesz rearrangement inequality

Schur test

Shapiro inequality

Sobolev inequality

Steffensen's inequality

Szegő inequality

Three spheres inequality

Trace inequalities

Trudinger's theorem

Turán's inequalities

Von Neumann's inequality

Wirtinger's inequality for functions

Inequalities relating to means

Hardy–Littlewood maximal inequality

Inequality of arithmetic and geometric means

Ky Fan inequality

Levinson's inequality

Maclaurin's inequality

Mahler's inequality

Muirhead's inequality

Newton's inequalities

Stein–Strömberg theorem

Young's inequality

Combinatorics

Binomial coefficient bounds

Factorial bounds

Fishburn–Shepp inequality

Fisher's inequality

Ingleton's inequality

Lubell–Yamamoto–Meshalkin inequality

Nesbitt's inequality

Rearrangement inequality

Schur's inequality

Stirling's formula (bounds)

Differential equations

Grönwall's inequality

Geometry

Alexandrov–Fenchel inequality

Aristarchus' inequality

Barrow's inequality

Berger–Kazdan comparison theorem

Blaschke–Santaló inequality

Bishop–Gromov inequality

Bogomolov–Miyaoka–Yau inequality

Bonnesen's inequality

Brascamp–Lieb inequality

Brunn–Minkowski inequality

Castelnuovo–Severi inequality

Cheng's eigenvalue comparison theorem

Clifford's theorem on special divisors

Cohn-Vossen's inequality

Erdős–Mordell inequality

Euler's theorem in geometry

Gromov's inequality for complex projective space

Gromov's systolic inequality for essential manifolds

Hadamard's inequality

Hadwiger–Finsler inequality

Hinge theorem

Hitchin–Thorpe inequality

Isoperimetric inequality

Jordan's inequality

Jung's theorem

Loewner's torus inequality

Łojasiewicz inequality

Loomis–Whitney inequality

Melchior's inequality

Milman's reverse Brunn–Minkowski inequality

Minkowski's first inequality for convex bodies

Myers's theorem

Noether inequality

Ono's inequality

Pedoe's inequality

Ptolemy's inequality

Pu's inequality

Riemannian Penrose inequality

Toponogov's theorem

Triangle inequality

Weitzenböck's inequality

Wirtinger inequality (2-forms)

Information theory

Inequalities in information theory

Kraft's inequality

Log sum inequality

Welch bounds

Algebra

Abhyankar's inequality

Pisier–Ringrose inequality

Linear algebra

Abel's inequality

Bregman–Minc inequality

Cauchy–Schwarz inequality

Golden–Thompson inequality

Linear matrix inequality

Peetre's inequality

Triangle inequality

von Neumann's trace inequality

Weyl's inequality

Number theory

Bonse's inequality

Large sieve inequality

Pólya–Vinogradov inequality

Turán–Kubilius inequality

Weyl's inequality

Probability theory and statistics

Azuma's inequality

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

Burkholder's inequality

Burkholder–Davis–Gundy inequalities

Cantelli's inequality

Chebyshev's inequality

Chernoff's inequality

Chung–Erdős inequality

Concentration inequality

Cramér–Rao inequality

Doob's martingale inequality

Dvoretzky–Kiefer–Wolfowitz inequality

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

Gauss's inequality

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

Gaussian isoperimetric inequality

Gibbs' inequality

Hoeffding's inequality

Hoeffding's lemma

Jensen's inequality

Khintchine inequality

Kolmogorov's inequality

Kunita–Watanabe inequality

Le Cam's theorem

Marcinkiewicz–Zygmund inequality

Markov's inequality

McDiarmid's inequality

Paley–Zygmund inequality

Pinsker's inequality

Popoviciu's inequality on variances

Rao–Blackwell theorem

Ross's conjecture, a lower bound on the average waiting time in certain queues

Samuelson's inequality

Shearer's inequality

Simpson's paradox

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

Clausius–Duhem inequality

Correlation inequality – any of several inequalities

FKG inequality

Ginibre inequality

Griffiths inequality

Heisenberg's inequality

Holley inequality

Leggett–Garg inequality

Riemannian Penrose inequality

Rushbrooke inequality

Sakurai's Bell inequality

Tsirelson's inequality

Wigner–d'Espagnat inequality