Wiktionary inequalityInequality may refer to In mathematics Inequality mathematics Inequalities book Inequalities book 1934 , a mathematics book by G. H. Hardy , J. E. Littlewood , and George Polya G. Polya In healthcare Health disparities Healthcare inequality In economics Economic inequality Income inequality metrics International inequality Income inequality in the United States Wealth inequality in the United States In the social sciences Educational inequality Gender inequality Participation inequality Social inequality Social equality Social stratification disambig da Ulighed es Desigualdad eo Neegala o fr In galit sh Nejednakost razvrstavanje ... more details
Wirtinger s inequality is either of two inequality mathematics inequalities named after Wilhelm Wirtinger Wirtinger s inequality for functions Wirtinger inequality 2 forms mathdab ... more details
In mathematics , Berger inequality may refer to Berger s inequality for Einstein manifolds the Berger&ndash Kazdan comparison theorem . mathdab ... more details
The following pages deal with inequalities due to Mikhail Gromov mathematician Mikhail Gromov Bishop&ndash GromovinequalityGromov s inequality for complex projective space Gromov s systolic inequality for essential manifolds L vy&ndash Gromovinequality disambig ... more details
seealso Social equality Horizontal inequality is the inequality economical, social or other that does not follow from a difference in an inherent quality such as intelligence, attractiveness or skills for people or profitability for corporations. In sociology, this is particularly applicable to forced inequality between different subcultures living in the same society. In economics, horizontal inequality is seen when people of similar origin, intelligence, etc. still do not have equal success and have different status, income and wealth. Traditional economic theory predicts that horizontal inequality should not exist in a free market. However, horizontal inequality is observed in real and simulated free market systems. The Pareto optimal economy is one traditional approach to the problem. Even in simulated systems, inequality of perfectly identical actors arises, to give the rich and poor . ref Eric Beinhocker. The Origin of Wealth Evolution, Complexity, and the Radical Remaking of Economics. Harvard Business School Press, 2006. ref See also Vertical inequality References reflist Category Socioeconomics Category Income distribution Category Economic problems Category Sociological terms Category Social inequality econ problem stub ... more details
Information inequality may mean in statistics, the Cram r Rao bound , an inequality for the variance of an estimator based on the information in a sample in information theory, inequalities in information theory describes various inequalities specific to that context. in sociology, Information Inequality and Social Barriers also in sociology, information inequity disambig ... more details
In probability and statistics , a correlation inequality is one of a number of inequalities satisfied by the correlation function s of a model. Such inequalities are of particular use in statistical mechanics and in percolation theory . ref cite book mr 0421547 last Ginibre first J. chapter Correlation inequalities in statistical mechanics. title Mathematical aspects of statistical mechanics pages 27&ndash 45 year 1972 publisher Amer. Math. Soc. location Providence, R. I. ref Examples include Bell s inequality FKG inequality Griffiths inequality , and its generalisation, the Ginibre inequality References Reflist External links springer id c c110420 first P.C. last Fishburn Category Probabilistic inequalities Category Statistical inequalities Category Statistical mechanics Category Inequalities statistics stub ... more details
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 together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming . See vector space , inner product , and normed vector space for other examples of how the basic ideas inherent in the triangle inequality line segment and distance can be generalized into a broader context. More formally, the Kantorovich inequality can be expressed this way Let math p i geq 0, quad 0 a leq x i leq b text for i 1, dots ,n. math Let math A n 1,2, dots ,n . math Then math begin align & qquad left sum i 1 n p ix i right left sum i 1 n frac p i x i right & leq frac a b 2 4ab left sum i 1 n p i right 2 frac a b 2 4ab cdot min left left sum i in X p i sum j in Y p j right 2 , , X cup Y A n , X cap Y varnothing right . end align math The Kantorovich inequality is used in convergence analysis it bounds the convergence rate of Cauchy s steepest descent . Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the Bunyakovsky inequality , the Wielandt inequality , and the Cauchy&ndash Schwarz inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the H lder inequality . The Kantorovich inequality is named after Soviet economist, mathematician, and Nobel Prize winner Leonid Kantorovich , a pioneer in the field of linear programming . References MathWorld urlname KantorovichInequality title Kantorovich Inequality PlanetMath urlname KantorovichInequality title Cauchy Schwarz inequality http carbon.cudenver.edu ... inequality External links http www groups.dcs.st and.ac.uk history Mathematicians Kantorovich.html ... more details
In mathematics, Peetre s inequality, named after Jaak Peetre , says that for any real number t and any Vector space vector s x and y in R sup n sup , the following inequality holds math left frac 1 x 2 1 y 2 right t le 2 t 1 x y 2 t . math References PlanetMath attribution id 4681 title Peetre s inequality mathanalysis stub Category Linear algebra Category Inequalities km ... more details
In mathematics , Steffensen s inequality , named after Johan Frederik Steffensen , is an integral inequality mathematics inequality in real analysis. It states that if a , b R is a non negative, monotonic function monotonically decreasing , integrable function and g a , b 0, 1 is another integrable function, then math int b k b f x , dx leq int a b f x g x , dx leq int a a k f x , dx, math where math k int a b g x , dx. math External links MathWorld title Steffensen s Inequality urlname SteffensensInequality Category Inequalities Category Real analysis mathanalysis stub km fi Steffensenin ep yht l ... more details
In the mathematical theory of Kleinian group s, J rgensen s inequality is an inequality involving the traces of elements of a Kleinian group , proved by harvs txt first Troels last J rgensen authorlink Troels J rgensen year 1976 . The inequality states that if A and B generate a non elementary discrete subgroup of SL sub 2 sub C , then math text Tr A 2 4 text Tr ABA 1 B 1 2 ge 1. , math References Citation last1 J rgensen first1 Troels title On discrete groups of M bius transformations jstor 2373814 mr 0427627 year 1976 journal American Journal of Mathematics issn 0002 9327 volume 98 issue 3 pages 739 749 DEFAULTSORT Jorgensen s Inequality Category Kleinian groups ... more details
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 inequality . More precisely, consider a planar simple closed curve of length math L math bounding a domain of area math F math . Let math r math and math R math denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality math L 2 4 pi F geq pi 2 R r 2. , math The term math pi 2 R r 2 math in the right hand side is known as the em isoperimetric defect em . Loewner s torus inequality with isosystolic defect is a Systolic geometry systolic analogue of Bonnesen s inequality. References Bonnesen, T. Sur une am lioration de l in galit isop rimetrique du cercle et la d monstration d une in galit de Minkowski, C. R. Acad. Sci. Paris 172 1921 , 1087 1089. Yu. D. Burago and V. A. Zalgaller, Geometric inequalities . Translated from the Russian by A. B. Sosinski . Springer Verlag, Berlin, 1988. ISBN 3 540 13615 0. Category Elementary geometry Category Geometric inequalities ... more details
Housing inequality refers to the differences in the quality of house housing that exist within a given ... on the negative aspects of Social inequalityinequality . ref Sen 2004 p. 61 ref The term may apply ... between groups of varying racial or social backgrounds. ref Pryce 2009 p. 145 ref Housing inequality is directly related to concepts of Racism in the United States racial inequality , Social inequality , Income inequality in the United States income inequality , and Wealth inequality in the United States wealth inequality . In addition, it is the result of a number of different factors including natural Free market market forces , Housing discrimination , and Housing Segregation . Housing inequality ... and an effect of poverty. ref Yinger 2001p. 360 ref Residential inequality is especially relevant ... of basic capabilities . ref Sen 1999 p. 87 ref Relation to economic inequality Housing inequality is a type of Economic inequality . This is due to the fact that disparities in housing explain variations ... John Yinger ref Yinger 2001 ref explains urban residential inequality as a result of natural housing ... to other socio economic factors as no single cause can explain housing inequality. In the United ... effect of residential inequality is an inequality of neighborhood amenities. Neighborhood amenities .... This furthers the negative effects of housing inequality by restricting access to Personal finance household wealth . ref Krivo and Kaufman 2004 ref The effects of housing inequality are necessarily related to economic inequality as they greatly affect the freedoms available to an individual. Proposed remedies There have been a number of plans proposed to remedy the negative effects of housing inequality ... in the United States Scattered site housing International housing inequality While the focus of housing inequality has changed over time, contemporary international analyses tend to center on Urbanization and the move to metropolitan areas. International housing inequality is largely characterized ... more details
Inequality Reexamined is a book by Amartya Sen first published by Harvard University Press . In it Sen evaluates the different perspectives of the general notion of inequality, focusing mainly on his well known capability approach . The author argues that inequality is a central notion to every social theory that has stood on time. For only if this basic feature is satisfied can a social theory which advocates a set of social arrangements be plausible. Taken the inequality ingredient for granted, the crucial question becomes inequality of what? Sen answers this basic question by advocating his preferred notion of equality which is based on the capability for functionings. Functionings and the role of freedom Confusing date January 2008 Functionings are of two kinds elementary ones such as being in good health, nourished, sheltered and the more complex, social ones such as having self respect, taking part in the life of the community etc. Achievement of an individual is the set of these realized functionings. Whereas capability refers to the real options that someone has in order to pursue his subjective functionings who prefers most. Nevertheless inequalities related to class, gender, communities seem to hinder the extent of human freedom and thus decrease our capability to function. That is why a good society ought to mitigate such discrimination, promoting people s freedom which, according to Sen, is the most valuable element of a satisfactory life. Editions cite book author Amartya Sen title Inequality Reexamined publisher Harvard University Press year 1992 isbn 0 674 45255 0 First Hardcover cite book author Amartya Sen title Inequality Reexamined publisher Harvard University Press year 1995 isbn 0 674 45256 9 Paperback Reprint External links Online version http www.oxfordscholarship.com oso public content economicsfinance 0198289286 toc.html Inequality Reexamined , Oxford Scholarship Online Category Philosophy books Category Sociology books sociology book stub ... more details
This disambiguation page had piped links removed by a bot, per WP MOSDAB . If some links do not seem to belong on this page, they may have originally been piped, so please check before removing links. Thanks In mathematics, Bernstein inequality may refer to Bernstein s inequality mathematical analysis Bernstein inequalities probability theory disambig Category Mathematical disambiguation ... more details
In probability theory , Etemadi s inequality is a so called maximal inequality , an inequality mathematics inequality that gives a bound on the probability that the partial sum s of a Finite set finite collection of independent random variables exceed some specified bound. The result is due to Nasrollah Etemadi . Statement of the inequality Let X sub 1 sub , ..., X sub n sub be independent real valued random variables defined on some common probability space , and let 0. Let S sub k sub denote the partial sum math S k X 1 cdots X k . , math Then math mathbb P left max 1 leq k leq n S k geq alpha right leq n max 1 leq k leq n mathbb P left S k geq alpha n right . math Remark Suppose that the random variables X sub k sub have common expected value zero. Apply Chebyshev s inequality to the right hand side of Etemadi s inequality and replace by 3. The result is Kolmogorov s inequality with an extra factor of 27 on the right hand side math mathbb P left max 1 leq k leq n S k geq alpha right leq frac 27 alpha 2 mathrm Var S n . math References cite book last Billingsley first Patrick title Probability and Measure publisher John Wiley & Sons, Inc. location New York year 1995 isbn 0 471 00710 2 Theorem 22.5 cite journal last Etemadi first Nasrollah title On some classical results in probability theory journal Sankhy Ser. A volume 47 year 1985 pages 215&ndash 221 mr 0844022 jstor 25050536 issue 2 Category Probabilistic inequalities Category Statistical inequalities ... more details
Eilenberg s inequality is a inequality mathematics mathematical inequality for Lipschitz continuity Lipschitz continuous function s. Let &fnof X Y be a Lipschitz continuous function between separable space separable metric space s whose Lipschitz constant is denoted by Lip &fnof . Then, Eilenberg s inequality states that math int Y H m n A cap f 1 y , dH n y leq frac v m n v n v m text Lip f n H m A , math for any A X and all 0 n m , where the asterisk denotes the upper Lebesgue integral , v sub n sub is the volume of the unit ball in R sup n sup , H sub n sub is the n dimensional Hausdorff measure . References Yu. D. Burago and V. A. Zalgaller, Geometric inequalities . Translated from the Russian by A. B. Sosinski . Springer Verlag, Berlin, 1988. ISBN 3 540 13615 0. Category Inequalities ... more details
Orphan date October 2011 In mathematics , a componentwise inequality mathematics inequality is an expression of the form math x,y in real n x preceq y iff x i leq y i forall i 1, ldots,n math ref http www.ee.ucla.edu ee236a lectures lineqs.pdf ref ref http www.stanford.edu class ee364a lectures sets.pdf ref ref http www.ece.ucsb.edu roy classnotes 271a ECE271a lecture3 small.pdf ref The Euclidean vector vector s do not have to be real number real , they can be from any space in which the inequality relation is defined. See also Pointwise Pointwise relations References Reflist Category Inequalities algebra stub ... more details
Mathanalysis stub In functional analysis , a mathematical discipline, the Szeg inequality or P lya&ndash Szeg inequality , named after George P lya and G bor Szeg , states that if math 1 leq p infty math and math u mathbb R n rightarrow mathbb R text in W 1,p mathbb R n , math then math int mathbb R n nabla u p , d mathcal H n leq int mathbb R n nabla u p , d mathcal H n. math See also G bor Szeg H lder s inequality Category Sobolev spaces it Disuguaglianza di Polya Szego ... more details
Abhyankar s inequality is an inequality involving extensions of valued field s in algebra , introduced by harvs txt authorlink Shreeram Shankar Abhyankar last Abhyankar year 1956 . If K k is an extension of valued field s, then Abhyankar s inequality states that the transcendence degree of K k is at least the transcendence degree of the residue field extension plus the Q rank Q rank of the quotient of the valuation group s. References Citation last1 Abhyankar first1 Shreeram title On the valuations centered in a local domain jstor 2372519 mr 0082477 year 1956 journal American Journal of Mathematics issn 0002 9327 volume 78 pages 321 348 Category Field theory Category Commutative algebra ... more details
In mathematics , Levinson s inequality is the following inequality, due to Norman Levinson , involving positive numbers. Let math a 0 math and let math f math be a given function having a third derivative on the range math 0,2a math , and such that math f x geq 0 math for all math x in 0,2a math . Suppose math 0 x i leq a math for math i 1, ldots, n math and math 0 p math . Then math frac sum i 1 np i f x i sum i 1 np i f left frac sum i 1 np ix i sum i 1 np i right le frac sum i 1 np if 2a x i sum i 1 np i f left frac sum i 1 np i 2a x i sum i 1 np i right . math The Ky Fan inequality is the special case of Levinson s inequality where math p i 1, a frac 1 2 , math and math f x log x. , math References Scott Lawrence and Daniel Segalman A generalization of two inequalities involving means , Proceedings of the American Mathematical Society. Vol 35 No. 1, September 1972. Norman Levinson Generalization of an inequality of Ky Fan , Journal of Mathematical Analysis and Applications. Vol 8 1964 , 133 134. Category Inequalities fi Levinsonin ep yht l ... more details
In mathematics , Ono s inequality is a theorem about triangle s in the Euclidean plane . In its original form, as conjecture d by T. Ono in 1914, the inequality is actually false however, the statement is true for acute triangle s, as shown by Balitrand in 1916. Statement of the inequality Consider an triangle Types of triangles acute triangle in the Euclidean plane with side lengths a , b and c and area A . Then math 27 b 2 c 2 a 2 2 c 2 a 2 b 2 2 a 2 b 2 c 2 2 leq 4 A 6. math This inequality fails for general triangles which was Ono s original conjecture , as shown by the counterexample a 3 4, b 1 2, c 1. External links MathWorld urlname OnoInequality title Ono inequality References cite journal last Balitrand first F. title Problem 4417 journal Intermed. Math. volume 23 pages 86&ndash 87 year 1916 cite journal last Ono first T. title Problem 4417 journal Intermed. Math. volume 21 pages 146 year 1914 cite journal last Quijano first G. title Problem 4417 journal Intermed. Math. volume 22 pages 66 year 1915 Category Disproved conjectures Category Triangle geometry Category Geometric inequalities it Disuguaglianza di Ono km sr ... more details
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 &fnof , and A , B , and C are the lengths of the sides of a triangle with area F , then math A 2 b 2 c 2 a 2 B 2 a 2 c 2 b 2 C 2 a 2 b 2 c 2 geq 16Ff, , math with equality if and only if the two triangles are similarity geometry similar . The expression on the left is not only symmetric under any of the six permutations of the set A , a , B , b , C , c of pairs, but also&mdash perhaps not so obviously&mdash remains the same if a is interchanged with A and b with B and c with C . In other words, it is a symmetric function of the pair of triangles. Pedoe s inequality is a generalization of Weitzenb ck s inequality and of the Hadwiger Finsler inequality . References A Two Triangle Inequality , Daniel Pedoe , The American Mathematical Monthly , volume 70, number 9, page 1012, November, 1963. An Inequality for Two Triangles , D. Pedoe, Proceedings of the Cambridge Philosophical Society , volume 38, part 4, page 397, 1943. External links http www.ele math.com files mia 07 2 full mia 07 32.pdf Pedoe s inequality Category Geometric inequalities Category Triangle geometry ar bs Pedoeova nejednakost de Ungleichung von Pedoe ko it Disuguaglianza di Pedoe nl Ongelijkheid van Pedoe km ru fi Pedoen ep yht l zh yue zh ... more details
Carleman s inequality is an inequality mathematics inequality in mathematics , named after Torsten Carleman , who proved it in 1923 ref T. Carleman, Sur les fonctions quasi analytiques , Conf rences faites au cinqui me congres des math maticiens Scandinaves, Helsinki 1923 , 181 196. ref and used it to prove the Denjoy&ndash Carleman theorem on quasi analytic classes. ref cite journal mr 2040885 last1 Duncan first1 John last2 McGregor first2 Colin M. title Carleman s inequality journal Amer. Math. Monthly volume 110 year 2003 issue 5 pages 424&ndash 431 ref ref cite journal mr 1820809 last1 Pe ari first1 Josip last2 Stolarsky first2 Kenneth B. title Carleman s inequality history and new generalizations journal Aequationes Math. volume 61 year 2001 issue 1&ndash 2 pages 49&ndash 62 ref Statement Let a sub 1 sub , a sub 2 sub , a sub 3 sub , ... be a sequence of non negative real number s, then math ... mathematical constant e in the inequality is optimal, that is, the inequality does not always hold if e is replaced by a smaller number. The inequality is strict it holds with < instead of &le if some element in the sequence is non zero. Integral version Carleman s inequality has an integral version ... f x dx math for any f 0. Carleson s inequality A generalisation, due to Lennart Carleson , states the following ref cite journal first L. last Carleson title A proof of an inequality of Carleman ... p e g x x dx leq e p 1 int 0 infty x p e g x dx. , math Carleman s inequality follows from the case p 0. Proof A short and elementary proof is available, and we sketch it here. From the inequality ... of inequality, math n ge sqrt 2 pi n , n n e n math , written for math n 1 math , implies math ... 1 n n 1 bigg , k a k , e , sum k ge1 , a k , , math proving the inequality. Moreover, the inequality ... math for math k 1, dots,n math . As a first consequence, Carleman s inequality is never an equality ... s inequality by starting with Hardy s inequality math sum n 1 infty left frac a 1 a 2 cdots a n n right ... more details
In mathematics , Schur s inequality mathematics inequality , named after Issai Schur , establishes that for all Nonnegative number non negative real number s x , y , z and a positive number t , math x t x y x z y t y z y x z t z x z y ge 0 math with equality if and only if x y z or two of them are equal and the other is zero. When t is an even positive integer , the inequality holds for all real numbers x , y and z . When math t 1 math , the following well known special case can be derived math x 3 y 3 z 3 3xyz geq xy x y xz x z yz y z math Proof Since the inequality is symmetric in math x,y,z math we may assume without loss of generality that math x geq y geq z math . Then the inequality math x y x t x z y t y z z t x z y z geq 0 , math clearly holds, since every term on the left hand side of the equation is non negative. This rearranges to Schur s inequality. Extension A generalization of Schur s inequality is the following Suppose a,b,c are positive real numbers. If the triples a,b,c and x,y,z are Order isomorphic similarly sorted , then the following inequality holds math a x y x z b y z y x c z x z y ge 0. math In 2007, Romania n mathematician Valentin Vornicu showed that a yet further generalized form of Schur s inequality holds Consider math a,b,c,x,y,z in mathbb R math , where math a geq b geq c math , and either math x geq y geq z math or math z geq y geq x math . Let math k in mathbb Z math , and let math f mathbb R rightarrow mathbb R 0 math be either convex function convex or monotonic . Then, math f x a b k a c k f y b a k b c k f z c a k c b k geq 0 . , math The standard form of Schur s is the case of this inequality where x a , y b , z c , k 1, m m sup r sup . ref Vornicu, Valentin Olimpiada de Matematica... de la provocare la experienta GIL Publishing House Zalau, Romania. ref Notes reflist Category Inequalities Category Articles containing proofs fr In galit de Schur ko it Disuguaglianza di Schur ja km vi B t ng th c ... more details
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