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
Image LabeledTriangle.svg thumb right 220px According to Weitzenb ck s inequality, the area of this triangle is at most math a sup 2 sup b sup 2 sup c sup 2 sup 4 overline 3 . In mathematics , Weitzenb ck s inequality named after Roland Weitzenb ck states that for a triangle of side lengths math a math , math b math , math c math , and area math Delta math , the following inequality holds math a 2 b 2 c 2 geq 4 sqrt 3 , Delta. math Equality occurs if and only if the triangle is equilateral. Pedoe s inequality is a generalization of Weitzenb ck s inequality. Proofs The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. Even so, the result is not too difficult to derive using Heron s formula for the area of a triangle math begin align Delta & frac sqrt a b c a b c b c a c a b 4 & frac 1 4 sqrt 2 a 2 b 2 a 2c 2 b 2c 2 a 4 b 4 c 4 . end align math First method This method assumes no knowledge of inequalities except that all squares are nonnegative. math begin align & a 2 b 2 2 b 2 c 2 2 c 2 a 2 2 geq 0 iff & 2 a 4 b 4 c 4 2 a 2 b 2 a 2c 2 b 2c 2 geq 0 iff & frac 4 a 4 b 4 c 4 3 geq frac 4 a 2 b 2 a 2c 2 b 2c 2 3 iff & frac a 4 b 4 c 4 2 a 2 b 2 a 2c 2 b 2c 2 3 geq 2 a 2 b 2 a 2c 2 b 2c 2 a 4 b 4 c 4 iff & frac a 2 b 2 c 2 2 3 geq 4 Delta .... From the first inequality we can also see that equality occurs only when math a b c math and the triangle is equilateral. Second method This proof assumes knowledge of the rearrangement inequality and the arithmetic geometric mean inequality . math begin align & & a 2 b 2 c 2 & geq & & ab bc ca ... sqrt3 Delta. end align math As we have used the rearrangement inequality and the arithmetic geometric mean inequality, equality only occurs when math a b c math and the triangle is equilateral. Third ... title Weitzenb ck s Inequality http demonstrations.wolfram.com WeitzenboecksInequality Weitzenb ck s Inequality , an interactive demonstration by Jay Warendorff Wolfram Demonstrations ... more details
. Isoperimetric inequality Pu s inequality bears a curious resemblance to the classical isoperimetry isoperimetric inequality math L 2 geq 4 pi A math for Jordan curve theorem Jordan curves in the plane ... quantity length . However, the inequality goes in the opposite direction. Thus, Pu s inequality can be thought of as an opposite isoperimetric inequality. See also Filling area conjecture Gromov s systolic inequality for essential manifolds Gromov s inequality for complex projective space Loewner s torus inequality Systolic geometry Systoles of surfaces References Mikhail Gromov mathematician Gromov ... more details
Hardy s inequality is an inequality mathematics inequality in mathematics , named after G. H. Hardy . It states that if math a 1, a 2, a 3, dots math is a sequence of non negative real number s which is not identically zero, then for every real number p 1 one has math sum n 1 infty left frac a 1 a 2 cdots a n n right p left frac p p 1 right p sum n 1 infty a n p. math An integral version of Hardy s inequality states if f is an integrable function with non negative values, then math int 0 infty left frac 1 x int 0 x f t , dt right p , dx le left frac p p 1 right p int 0 infty f x p , dx. math Equality holds if and only if f x 0 almost everywhere . Hardy s inequality was first published without proof in 1920 in a note by Hardy. ref Hardy, G.H., Note on a Theorem of Hilbert , Math. Z. 6 1920 , 314&ndash 317. ref The original formulation was in an integral form slightly different from the above. See also Carleman s inequality Notes references References cite book last Hardy first G. H. coauthors Littlewood. J.E. P lya, G. title Inequalities, 2nd ed publisher Cambridge University Press year 1952 pages isbn 0521358809 cite book last Kufner first Alois coauthors Persson, Lars Erik title Weighted inequalities of Hardy type publisher World Scientific Publishing year 2003 pages isbn 9812381953 Citation last1 Ribari first1 M. title On some inequalities for convex functions year 1973 journal Mathematica Balkanica volume 3 pages 435 442. Category Inequalities Category Real analysis bs Hardyjeva nejednakost ko hu Hardy egyenl tlens g km fi Hardyn ep yht l sv Hardys olikhet ... more details
File Jordans inequality.svg thumb upright 1.2 unit circle with angle x and a second circle with radius math EG sin x math around E. math begin align & DE leq widehat DC leq widehat DG Leftrightarrow & sin x leq x leq tfrac pi 2 sin x Rightarrow & tfrac 2 pi x leq sin x leq x end align math In mathematics, Jordan s inequality , named after Camille Jordan , states that math frac 2 pi x leq sin x leq x text for x in left 0, frac pi 2 right . math It can be proven through the geometry of circles see drawing . ref http planetmath.org encyclopedia ProofOfJordansInequality.html ref Notes references Further reading Serge Colombo Holomorphic Functions of One Variable . Taylor & Francis 1983, ISBN 0677059507, p. 167 168 http books.google.de books?id pFEOAAAAQAAJ&pg PA167 online copy Da Wei Niu, Jian Cao, Feng Qi http www.scientificbulletin.upb.ro rev docs arhiva full3105.pdf Generealizations of Jordan s Inequality and Concerned Relations . U.P.B. Sci. Bull., Series A, Volime 72, Issue 3, 2010, ISSN 1223 7027 Peng Qi http www.ajmaa.org RGMIA papers v9n3 refine jordan kober.pdf Jordan s Inequality Refinements, Generealizations, Applications and related Problems . RGMIA Res Rep Coll 2006 , Volume 9, Issue 3, Pages 243 259 Meng Kuang Kuo http www.journalofinequalitiesandapplications.com content 2011 1 130 Refinements of Jordan s inequality . Journal of Inequalities and Applications 2011, 2011 130, doi 10.1186 1029 242X 2011 130 External links http planetmath.org encyclopedia JordansInequality.html Jordan s inequality at PlanetMath MathWorld title Jordan s inequality urlname JordansInequality Category Inequalities Mathanalysis stub de Jordan Ungleichung ... more details
Image Barrow inequality.svg thumb right 300px 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 bisector s of BPC , CPA , and APB intersect the sides BC , CA , AB , respectively. Then math PA PB PC geq 2 PU PV PW , , math with equality holding only in the case of an equilateral triangle . Barrow s inequality strengthens the Erd s Mordell inequality , which has identical form except with PU , PV , and PW replaced by the three distances of P from the triangle s sides. It is named after David Francis Barrow. See also Euler s theorem in geometry External links http www.eleves.ens.fr home kortchem olympiades Cours Inegalites tin2006.pdf Hojoo Lee Topics in Inequalities Theorems and Techniques Category Triangle geometry Category Geometric inequalities Category Article Feedback 5 km pl Nier wno Barrowa ro Inegalitatea lui Barrow fi Barrow n ep yht l ... more details
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. Let f sub n sub be a sequence of real number s such that f sub n sub f sub n 1 sub 0 for n 1, 2, , and let a sub n sub be a sequence of real or complex number s. Then math left sum n 1 m a n f n right le A m f 1, math where math A m operatorname max left lbrace a 1 , a 1 a 2 , dots, a 1 a 2 cdots a m right rbrace. math The inequality also holds for infinite series , in the limit as math m rightarrow infty math , if math lim m rightarrow infty A m math exists. References mathworld title Abel s inequality urlname AbelsInequality Category Inequalities Category Niels Henrik Abel mathanalysis stub bs Abelova nejednakost fa ro Inegalitatea lui Abel ... more details
In information theory , Pinsker s inequality , named after its inventor Mark Semenovich Pinsker , is an inequality mathematics inequality that relates Kullback Leibler divergence and the total variation distance . It states that if P , Q are two probability distribution s, then math sqrt D P Q 2 ge sup P A Q A A text is an event to which probabilities are assigned. math where D P Q is the Kullback Leibler divergence in Nat information nats and math sup A P A Q A , math is the Total variation distance of probability measures total variation distance between P and Q . References Thomas M. Cover and Joy A. Thomas Elements of Information Theory , 2nd edition, Willey Interscience, 2006 Nicolo Cesa Bianchi and G bor Lugosi Prediction, Learning, and Games , Cambridge University Press, 2006 Category Information theory Category Probabilistic inequalities vi B t ng th c Pinsker ... more details
In mathematics , especially functional analysis , Bessel s inequality is a statement about the coefficients of an element math x math in a Hilbert space with respect to an orthonormal sequence . Let math H math be a Hilbert space, and suppose that math e 1, e 2, ... math is an orthonormal sequence in math H math . Then, for any math x math in math H math one has math sum k 1 infty left vert left langle x,e k right rangle right vert 2 le left Vert x right Vert 2 math where , denotes the inner product space inner product in the Hilbert space math H math . If we define the infinite sum math x sum k 1 infty left langle x,e k right rangle e k, math consisting of infinite sum of vector resolute math x math in direction math e k math , Bessel s inequality mathematics inequality tells us that this series mathematics series Limit of a sequence converges . One can think of it that there exists math x in H math which can be described in terms of potential basis math e 1, e 2, ... math . For a complete orthonormal sequence that is, for an orthonormal sequence which is a Orthonormal basis basis , we have Parseval s identity , which replaces the inequality with an equality and consequently math x math with math x math . Bessel s inequality follows from the identity math 0 le left x sum k 1 n langle x, e k rangle e k right 2 x 2 2 sum k 1 n langle x, e k rangle 2 sum k 1 n langle x, e k rangle 2 x 2 sum k 1 n langle x, e k rangle 2, math which holds for any natural n . See also Cauchy Schwarz inequality External links http mathworld.wolfram.com BesselsInequality.html Bessel s Inequality the article on Bessel s Inequality on MathWorld. PlanetMath attribution title Bessel inequality id 3089 Category Hilbert space Category Inequalities ca Desigualtat de Bessel de Besselsche Ungleichung es Desigualdad de Bessel eo Neegala o de Bessel fr In galit de Bessel it Disuguaglianza di Bessel kk hu Bessel egyenl tlens g ro Inegalitatea lui Bessel ru fi Besselin ... more details
In mathematics , Nesbitt s inequality mathematics inequality is a special case of the Shapiro inequality . It states that for positive real numbers a , b and c we have math frac a b c frac b a c frac c a b geq frac 3 2 . math Proof First proof Starting from Nesbitt s inequality 1903 math frac a b c frac b a c frac c a b geq frac 3 2 math we transform the left hand side math frac a b c b c frac a b c a c frac a b c a b 3 geq frac 3 2 . math Now this can be transformed into math a b a c b c left frac 1 a b frac 1 a c frac 1 b c right geq 9. math Division by 3 and the right factor yields math frac a b a c b c 3 geq frac 3 frac 1 a b frac 1 a c frac 1 b c . math Now on the left we have the arithmetic mean and on the right the harmonic mean , so this inequality is true. We might also want to try to use GM for three variables. Second proof Suppose math a ge b ge c math , we have that math frac 1 b c ge frac 1 a c ge frac 1 a b math define math vec x a, b, c math math vec y frac 1 b c , frac 1 a c , frac 1 a b math The scalar product of the two sequences is maximum because of the Rearrangement inequality if they are arranged the same way, call math vec y 1 math and math vec y 2 math the vector ... math math vec x cdot vec y ge vec x cdot vec y 2 math Addition yields Nesbitt s inequality. Third proof ... inequality can be solved by transforming it to the appropriate identity, see Hilbert s seventeenth problem . Fourth proof Starting from Nesbitt s inequality 1903 math frac a b c frac b a c frac ... 1 b c frac 1 a c frac 1 a b right geq 9 math Which is true by the Cauchy Schwarz inequality . Fifth proof Starting from Nesbitt s inequality 1903 math frac a b c frac b a c frac c a b geq frac 3 2 math ... is true, by inequality of arithmetic and geometric means . References cite web title Introduction ... for more proofs of this inequality. PlanetMath reference id 2875 title Nesbitt s inequality PlanetMath reference id 2876 title proof of Nesbitt s inequality DEFAULTSORT Nesbitt s Inequality Category ... more details
In mathematics , Maclaurin s inequality , named after Colin Maclaurin , is a refinement of the inequality of arithmetic and geometric means . Let a sub 1 sub , a sub 2 sub , ..., a sub n sub be positive number positive real number s, and for k 1, 2, ..., n define the averages S sub k sub as follows math S k frac displaystyle sum 1 leq i 1 cdots i k leq n a i 1 a i 2 cdots a i k displaystyle n choose k . math The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a sub 1 sub , a sub 2 sub , ..., a sub n sub , that is, the sum of all products of k of the numbers a sub 1 sub , a sub 2 sub , ..., a sub n sub with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient math scriptstyle n choose k . math Maclaurin s inequality is the following chain of inequalities math S 1 geq sqrt S 2 geq sqrt 3 S 3 geq cdots geq sqrt n S n math with equality if and only if all the a sub i sub are equal. For n 2, this gives the usual inequality of arithmetic and geometric means of two numbers. Maclaurin s inequality is well illustrated by the case n 4 math begin align & quad frac a 1 a 2 a 3 a 4 4 8pt & ge sqrt frac a 1a 2 a 1a 3 a 1a 4 a 2a 3 a 2a 4 a 3a 4 6 8pt & ge sqrt 3 frac a 1a 2a 3 a 1a 2a 4 a 1a 3a 4 a 2a 3a 4 4 8pt & ge sqrt 4 a 1a 2a 3a 4 . end align math Maclaurin s inequality can be proved using the Newton s inequalities . See also Newton s inequalities Muirhead s inequality Generalized mean inequality References cite book last Biler first Piotr coauthors Witkowski, Alfred title Problems in mathematical analysis publisher New York, N.Y. M. Dekker date 1990 pages isbn 0824783123 PlanetMath attribution id 3835 title MacLaurin s Inequality Category Real analysis Category Inequalities Category Symmetric functions ko it Disuguaglianza di MacLaurin hu MacLaurin egyenl tlens ge ... more details
hatnote This page is about Minkowski s inequality for norms. See Minkowski s first inequality for convex bodies for Minkowski s inequality in convex geometry. In mathematical analysis , the Minkowski inequality establishes that the Lp space L sup p sup spaces are normed vector space s. Let S be a measure space , let 1 p and let f and g be elements of L sup p sup S . Then f g is in L sup p sup S , and we have the triangle inequality math f g p le f p g p math with equality for 1 p < if and only ... supremum math f infty operatorname ess sup x in S f x . math The Minkowski inequality is the triangle inequality in L sup p sup S . In fact, it is a special case of the more general fact ... to see that the right hand side satisfies the triangular inequality. Like H lder s inequality , the Minkowski inequality can be specialized to sequences and vectors by using the counting measure ... s inequality , math left frac 1 2 a frac 1 2 b right p le frac 1 2 a p frac 1 2 b p. math This means ... p math . If it is zero, then Minkowski s inequality holds. We now assume that math f g p math is not zero. Using H lder s inequality math f g p p int f g p , mathrm d mu math math le int f g f g p 1 , mathrm ... p frac f g p p f g p . math We obtain Minkowski s inequality by multiplying both sides by math frac f g p f g p p . math Minkowski s integral inequality Suppose that S sub 1 sub , sub 1 sub and S sub ... s integral inequality is harv Stein 1970 loc A.1 , harv Hardy Littlewood P lya 1988 loc ... 1,2 , then Minkowski s integral inequality gives the usual Minkowski inequality as a special case for putting &fnof sub i sub y F i , y for i 1,2, the integral inequality gives math begin align f 1 ... F x,y p ,d mu 2 y right 1 p d mu 1 x & f 1 p f 2 p. end align math See also Mahler s inequality H lder s inequality References Cite book last1 Hardy first1 G. H. author1 link G. H. Hardy last2 Littlewood ... inequality author M.I. Voitsekhovskii cite web title Introduction to Inequalities url http www.mediafire.com ... more details
distinguish Popoviciu s inequality on variances In convex analysis , Popoviciu s inequality is an inequality mathematics inequality about convex function s. It is similar to Jensen s inequality and was found in 1965 by Tiberiu Popoviciu ref citation author Tiberiu Popoviciu title Sur certaines in galit s qui caract risent les fonctions convexes year 1965 journal Analele tiin ifice Univ. Al.I. Cuza Iasi, Sec ia I a Mat. volume 11 pages 155 164 ref , a Romanian mathematician. It states blockquote Let &fnof be a function from an interval math I subseteq mathbb R math to math mathbb R math . If &fnof is convex function convex , then for any three points math x, y, z math from math I math , math begin align & qquad frac f x f y f z 3 f left frac x y z 3 right 6pt & ge frac 2 3 left f left frac x y 2 right f left frac y z 2 right f left frac z x 2 right right . end align math If a function &fnof is continuous , then it is convex if and only if the above inequality holds for all x , y , z from math I math . When &fnof is strictly convex, the inequality is strict except for x y z . ref citation year 2006 title Convex functions and their applications a contemporary approach author1 Constantin Niculescu author2 Lars Erik Persson publisher Springer Science & Business isbn 9780387243009 page 12 url http books.google.com ?id M5tYCzB8FQcC&pg PA12&dq Popoviciu 27s inequality ref blockquote It can be generalised to any finite number n of points instead of 3, taken on the right hand side k at a time instead of 2 at a time ref citation year 1992 title Convex functions, partial orderings, and statistical applications author1 J. E. Pe ari author2 Frank Proschan author3 Yung Liang Tong publisher Academic Press isbn 9780125492508 page 171 url http ... s inequality can also be generalised to a weighted inequality. ref citation year 1976 author1 P. M ... Generalizations of Popoviciu s inequality class math.FA year 2008 accessdate 2009 08 26 ref Notes ... more details
merge to Occupational inequality date October 2011 Multiple issues essay like January 2011 wikify January 2011 globalize June 2011 Inequality in the workplace is any form of bias or discrimination that takes place in a work environment and is established, promoted, or allowed to persist by the workplace authority. It is common for a workplace to have inequalities based on gender , race classification of humans race , and social class . Citation needed date July 2011 There is usually a hierarchy that exists in the workplace in which managers, leaders and executives are paid higher wages and have more authority and prestige than those below them. ref Acker, Joan. Inequality Regimes . The Kaleidoscope of Gender, 2011, p. 355. ref Gender inequality Some common inequalities that take place in the workplace are the gender based imbalances of individuals in power philosophy power and command over the management of the organization. Women are not able to move up into higher paid positions quickly as compared to men. Some organizations have more inequality than others, and the extent to which it occurs can differ greatly. In the workplace the men usually hold the higher positions and the women often hold lower paid positions such as secretaries . ref Tomaskovic Devey, Donald. Gender & Racial Inequality at Work , 1993, p. 12. ref Racial inequality Ethnicity has a large influence on the quality of jobs as well as the amount of money an individual will make in the workforce. Today, African American men working full time and year round have 72 percent of the average earnings of comparable ... the most power are usually occupied by white men. Though this type of inequality has been lowered ... the Causes of Sexual Inequality. , 1978. ref In the workplace it is required that a worker .... This trend is part of what leads to modern day inequality. References Reflist Category Social inequality ... more details
For the similarly named inequality involving series Chebyshev s sum inequality In probability theory , Chebyshev s inequality also spelled as Tchebysheff s inequality guarantees that in any sample statistics ... than k standard deviations away from the mean. The inequality has great utility because it can ... Chebyshev s inequality may also refer to the Markov s inequality , especially in the context of analysis ... nowrap k 1 the right hand side is greater than one, so the inequality becomes vacuous, as the probability ... to completely arbitrary distributions unknown except for mean and variance , the inequality generally ..., by Chebyshev s inequality. But if we additionally know that the distribution is normal, we can ..., the bounds provided by Chebyshev s inequality cannot, in general remaining sound for variables of arbitrary ... Chebyshev inequality A one tailed variant with k 0, is ref Grimmett and Stirzaker, problem 7.11.9 ... math Pr X mu geq k sigma leq frac 1 1 k 2 . math The one sided version of the Chebyshev inequality is called Cantelli s inequality, and is due to Francesco Paolo Cantelli . An application distance .... But this inequality is trivially true if the variance is infinite. The proof is as follows. Setting k 1 in the statement for the one sided inequality gives math Pr X mu geq sigma leq frac ... Thus the median is within one standard deviation of the mean. For a proof using Jensen s inequality see Median An inequality relating means and medians . Proof of the two sided Chebyshev s inequality ... inequality follows from dividing the above inequality by g t . Probabilistic proof Markov s inequality states that for any real valued random variable Y and any positive number a , we have Pr Y a E Y a . One way to prove Chebyshev s inequality is to apply Markov s inequality .... In some cases it exceeds 1 by a very wide margin. Chebyshev s Inequality More General A more general version of Chebyshev s Inequality states math Pr X ge varepsilon le frac operatorname E X 2 varepsilon ... more details
Multiple issues tone April 2008 wikify January 2011 This article discusses social inequality income inequality in the United States in the United States and its effects on individual health , and more specifically likelihood of developing disease s. While incidence rate rates of incidence for many diseases vary based on biological factors and inheritable characteristics, a larger health disparities disparity , which cannot be explained by biological factors, exists in disease rates among varying racial and socioeconomic groups in the United States for example, poverty in the United States lower income African Americans and American upper class upper class Caucasian race Caucasians . This suggests that social and economic factors play a role in determining who acquires certain diseases in the United States. For example, heart disease is the most dangerous disease in America, followed closely by cancer , with the fifth most deadly being diabetes . The general risk factors associated with these three diseases include obesity and poor nutrition diet , tobacco and alcohol use, physical inactivity , and access to medical care and health information. Ref women While some of these risk factors are individual health choices, all of them are also correlated with socioeconomic factors, such as gender , Race classification of humans race , income , Environment systems environment , and education , and consequently, a person s likelihood for developing heart disease, cancer, or diabetes is in part correlated with these social factors. Men are more likely than women to die from heart disease. Likewise, African Americans and other Minorities in the united states racial minorities have higher ... inequality display higher rates of heart disease than populations with more evenly distributed ... 20043094886 title Income, income inequality, and cardiovascular disease mortality accessdate April 13 ... Inequality In Disease Category Social inequality ... more details
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