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# Topic: Inequality of arithmetic and geometric means

 PlanetMath: arithmetic-geometric-harmonic means inequality See Also: arithmetic mean, geometric mean, harmonic mean, general means inequality, weighted power mean, power mean, root-mean-square, proof of general means inequality, Jensen's inequality, derivation of geometric mean as the limit of the power mean, minimal and maximal number, proof of arithmetic-geometric means inequality using Lagrange multipliers inequality, mean, arithmetic mean, geometric mean, harmonic mean This is version 5 of arithmetic-geometric-harmonic means inequality, born on 2001-08-18, modified 2004-06-05. planetmath.org /encyclopedia/ArithmeticGeometricMeansInequality.html   (158 words)

 PlanetMath: proof of arithmetic-geometric-harmonic means inequality Now we shall prove the inequality between arithmetic mean and geometric mean. So far we have proved the inequality between the arithmetic mean and the geometric mean. This is version 3 of proof of arithmetic-geometric-harmonic means inequality, born on 2002-05-30, modified 2005-09-29. planetmath.org /encyclopedia/ProofOfArithmeticGeometricHarmonicMeansInequality.html   (228 words)

 Inequality of arithmetic and geometric means There is a similar inequality for the weighted arithmetic mean and weighted geometric mean. Other generalizations of the inequality of arithmetic and geometric means are given by Muirhead's inequality and Generalized mean inequality. The inequality of arithmetic and geometric means follows by straightforward algebraic manipulations. www.algebra.com /algebra/homework/Inequalities/Inequality_of_arithmetic_and_geometric_means.wikipedia   (842 words)

 Math 7030 The geometric-arithmetic mean inequality states for any set of positive numbers, the geometric mean is always less than or equal to the arithmetic mean and equality occurs if and only if all the numbers in the list are equal. On the left we have the geometric mean of our original set whereas on the right we have the arithmetic mean of the original set. The result on geometric and arithmetic means implies that the maximum area occurs when these numbers are the same. www.math.uga.edu /~john/geometry/7030_activity_10.htm   (1066 words)

 Arithmetic and geometric means (via CobWeb/3.1 planetlab2.cs.umd.edu)   (Site not responding. Last check: 2007-11-04) Arithmetic mean of the given numbers is defined as which actually says that the arithmetic mean has not been changed by addition of new terms. Q.E.D. There is a way to derive a complete proof of the inequality from the Pythagorean Theorem. www.cut-the-knot.org.cob-web.org:8888 /Generalization/means.shtml   (148 words)

 Geometric means, exponentials, and logs The expression I used for the geometric mean in question 8 of the sample quiz, for N=2, was sqrt(X1 * X2). Formula (3.4) also makes it clear how the "geometric mean" really is the *mean* of something; namely, first you take logarithms, then compute their mean, then "undo" the logarithm (that's the exponential). More to the point, we will use this inequality later to demonstrate that geometric means are "biased": on the average, they underestimate the true average concentrations or masses of chemicals in an environmental medium, for instance. www.quantdec.com /envstats/notes/class_03/gm_etc.htm   (963 words)

 J. Sandor On the Jensen-Hadamard inequality, Studia Univ.Babes-Bolyai, Mathematica, 36 (1991), 9-15. On certain identities for means, Studia Univ. Babes-Bolyai, 38 (1993), 7-14. On an inequality for the sum of infimums of functions (in coop. rgmia.vu.edu.au /sandor.html   (319 words)

 Highbeam Encyclopedia - Search Results for Arithmetic mean   (Site not responding. Last check: 2007-11-04) The arithmetic mean of a group of numbers is found by dividing their sum by the number of members in the group; e.g., the sum of the seven numbers 4, 5, 6, 9, 13, 14, and 19 is 70 so their mean is 70 divided by 7, or 10. An upper prediction limit for the arithmetic mean of a lognormal random variable. Ways of Means Committee: There's a critical difference between arithmetic and geometric means. www.encyclopedia.com /SearchResults.aspx?Q=Arithmetic+mean   (612 words)

 Generalized mean - Medbib.com, the modern encyclopedia (via CobWeb/3.1 planetlab2.cs.umd.edu)   (Site not responding. Last check: 2007-11-04) A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. , the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means. A power mean serves a non-linear moving average which is shifted towards small signal values for small p and emphasizes big signal values for big p. www.medbib.com.cob-web.org:8888 /Generalized_mean   (279 words)

 PlanetMath: proof of arithmetic-geometric means inequality using Lagrange multipliers (via CobWeb/3.1 ...   (Site not responding. Last check: 2007-11-04) This kind of symmetry argument was used to great effect by Jakob Steiner in his famous attempted proof of the isoperimetric inequality: among all closed (smooth) curves of a given arc length, the circle maximizes the "proof of arithmetic-geometric means inequality using Lagrange multipliers" is owned by stevecheng. This is version 12 of proof of arithmetic-geometric means inequality using Lagrange multipliers, born on 2005-07-27, modified 2006-10-07. planetmath.org.cob-web.org:8888 /encyclopedia/ProofOfArithmeticGeometricMeansInequalityUsingLagrangeMultipliers.html   (837 words)

 Question Corner -- Applications of the Geometric Mean It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20. If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%, and 20% would give you). The arithmetic mean can also be interpreted as the length of the sides of a square whose perimeter is the same as our rectangle. www.math.toronto.edu /mathnet/questionCorner/geomean.html   (2387 words)

 Homework 4.9 In this case, the geometric mean (9) is less than the arithmetic mean (15). That is, the arithmetic mean is greater than or equal to the geometric mean. That is, the geometric mean of three numbers gives you a side of a cube whose volume is the same as the volume of the rectangular box with the three given numbers as dimensions. www.math.fau.edu /Klingler/mfla1/hmwk4-9s.html   (548 words)

 v5n4 We consider certain refinements of the arithmetic and geometric means, the results generalize an inequality of P. Diananda. The sharpest bound is in terms of the one norm of the Appell polynomial which constitutes the coefficients of the derivative of the function to be approximated. Using sharp inequalities of trapezoid and midpoint type in terms of the infinum and supremum of the derivative, some new and better approximation of f-divergence are given. rgmia.vu.edu.au /v5n4.html   (629 words)

 v4n1 In this article, using Stirling's formula, the series-expansion of digamma functions and other techniques, two inequalities involving the geometric mean of natural numbers and the ratio of gamma functions are obtained. In this article, using inequality between logarithmic mean and one-parameter mean, which can be deduced from monotonicity of the extended mean values, an integral analogue of J.S. Martins' inequality is proved. Some inequalities related to the Ky Fan and C.-L. Wang inequalities for weighted arithmetic and geometric means are given. rgmia.vu.edu.au /v4n1.html   (791 words)

 FCPS Instructional Services: High School Instruction & K-12 Curriculum Services Graph the system of inequalities and identify the area of intersection as the feasible region. 8) Analyze data using measures of central tendency (mean, median, and mode), the range of the data, and box-and-whiskers plots. Students investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems. www.fcps.k12.va.us /DIS/OHSICS/math/algebra2h.htm   (1680 words)

 Monotonic Refinements of a Ky Fan Inequality   (Site not responding. Last check: 2007-11-04) It is well-known that inequalities between means play a very important role in many branches of mathematics. ALZER, Inequalities for arithmetic, geometric and harmonic means, Ky Fan inequality, Monotonic refinements of inequalities, Arithmetic, geometric and harmonic means. www.maths.tcd.ie /EMIS/journals/JIPAM/v2n2/035_00.html   (329 words)

 Homogeneous Functions   (Site not responding. Last check: 2007-11-04) Arithmetic mean, a(x, y, z,...) = (x + y + z +...)/N Harmonic mean, h(x, y, z,...) = N/(1/x + 1/y + 1/z +...) A famous inequality relates arithmetic and geometric means of nonnegative numbers: www.cut-the-knot.org /Generalization/homo.shtml   (317 words)

 Muirhead's Inequality   (Site not responding. Last check: 2007-11-04) Muirhead's Inequality is, in a sense, a sweeping generalization of the AM-GM inequality of n variables, with the symmetric mean [(1,0,...)] representing AM, and [(1/n,1/n,...)] representing GM. Andre Rzym: Muirhead's Inequality is a wonderfully well-written article giving background terminology and practical examples of how Muirhead's Inequality is used using the "majorize" approach. The AM-GM Inequality: the Arithmetic Mean of positive numbers is always greater than the Geometric Mean. mcraefamily.com /MathHelp/BasicNumberIneqMuirheadsInequality.htm   (1312 words)

 Muirhead's inequality - Wikipedia, the free encyclopedia In mathematics, Muirhead's inequality, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. In case a = (1/n,..., 1/n), it is the geometric mean of x Thus, a doubly stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each row and the sum of the entries in each column is 1. en.wikipedia.org /wiki/Muirhead's_inequality   (401 words)

 PlanetMath: (via CobWeb/3.1 planetlab2.cs.umd.edu)   (Site not responding. Last check: 2007-11-04) AGMH inequality (=arithmetic-geometric-harmonic means inequality) owned by drini AGM inequality (=arithmetic-geometric-harmonic means inequality) owned by drini arithmetic-geometric means inequality (=arithmetic-geometric-harmonic means inequality) owned by drini planetmath.org.cob-web.org:8888 /encyclopedia/A   (2476 words)

 Chapter3 The use of Cauchy's arithmetic-geometric inequality led to the name of geometric programming for the technique. This research went directly into the text Foundations of Optimization, which was published in 1967, and the result is that geometric programming is now applicable to a polynomial economic model with polynomial constraints as equalities and inequalities. However, it will require less effort to obtain the final result, since the background arguments associated with the geometric-arithmetic inequality will not be required, and the results for polynomial optimization will follow directly from posynomial optimization. www.mpri.lsu.edu /textbook/Chapter3.htm   (508 words)

 Publications An inequality for the chromatic number of graph (with George Szekeres), J. Some applications of the inequality of arithmetic and geometric means to polynomial equations, Proc. On Hilbert's inequality in n dimensions, (with N. de Bruijn), Bull. www.cis.upenn.edu /~wilf/reprints.html   (1280 words)

 KöMaL: English issue, December 2002 Using the notations of the figure, the area of the quadrilateral is From the inequality between the arithmetic and geometric means, If the corresponding inequality is set up for each face of the polyhedron, and the inequalities are summed, the left-hand side will be 4A, and the right-hand side will be 2Q, as each edge belongs to two faces of the convex polyhedron. www.komal.hu /lap/2002-ang/b3416.e.shtml   (109 words)

 Mathematics Magazine: April 2001 The Arithmetic-Geometric Mean Inequality guarantees that GM/AM cannot exceed 1. I show that there is a simple geometric interpretation of these terms as, respectively, the area between two curves and the area of a rectangle. The geometric picture, in turn, allows an informal derivation of the rule that should appeal to visually oriented students. www.maa.org /pubs/mag_apr01_toc.html   (863 words)

 Homework 4.9   (Site not responding. Last check: 2007-11-04) For each of the following pairs of numbers, find both the arithmetic mean and the geometric mean, and compare them. Show that the geometric mean of the numbers 10, 16, and 50 is less than their arithmetic mean. Give a geometric interpretation of the geometric mean of three numbers. www.math.fau.edu /Klingler/mfla1/hmwk4-9.html   (170 words)

 HKAL Pure Mathematics Arithmetic and Geometric Means Prove the well known inequality but just emphasize a fact that was used by Cauchy in his proof. Cauchy's Inequality (Eric Weisstein's World of Mathematics) Proof of the inequality by discriminant; vector approach of the inequality. Arithmetic Operations with Functions (Tools for Analyzing Functions) This tool is used to graph functions which are arithmetic combinations of two functions. ws1.hkcampus.net /~ws1-kcy/al_pmath.html   (1600 words)

 What makes e natural? Thus to the product of any two terms of the original sequence, there is a corresponding term of the arithmetic progression that is obtained by adding the terms assigned to the two terms multiplied. The convexity of the exponential function can be proved by means of the inequality of weighted arithmetic and geometric means. The monotonicity of each function also can be proved more directly using the inequality of weighted geometric and arithmetic means. www.komal.hu /cikkek/2004-ang/e.e.shtml   (1730 words)

 Dror Bar-Natan:Odds, Ends, Unfinished:Means Inequality There is a little known lovely geometric counterpart to the well known Arithmetic/Geometric Means Inequality. It implies the inequality of the means, of course, but it is not implied by it. After all, we are in the 21st century and its not so easy to come up with new proofs of high-school level inequalities. www.math.toronto.edu /~drorbn/projects/ArithGeom   (821 words)

 Publications of J. Sandor On the inequality of the arithmetic and geometric means, On certain inequalities for the distances of a point to the vertices and the sides of a triangle, Inequalities for certain means in two arguments (In coop.with.I.Rasa), sci.vu.edu.au /~rgmia/sandor.html   (265 words)

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