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An Elementary Proof of the Mean Inequalities

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An Elementary Proof of the Mean Inequalities Advances in Pure Mathematics, 2013, 3, 331-334 http://dx.doi.org/10.4236/apm.2013.33047 Published Online May 2013 (http://www.scirp.org/journal/apm) An Elementary Proof of the Mean Inequalities Ilhan M. Izmirli Department of Statistics, George Mason University, Fairfax, USA Email: [email protected] Received November 24, 2012; revised December 30, 2012; accepted February 3, 2013 Copyright © 2013 Ilhan M. Izmirli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the har- monic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and iden- tric means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem. Keywords: Pythagorean Means; Arithmetic Mean; Geometric Mean; Harmonic Mean; Identric Mean; Logarithmic Mean 1. Pythagorean Means 11 11 x HM HM x For a sequence of numbers x xx12,,, xn we will 12 let The geometric mean of two numbers x1 and x2 can n be visualized as the solution of the equation x j AM x,,, x x AM x j1 12 n x GM n 1 n GM x2 1 n GM x12,,, x xnj GM x x j1 1) GM AM HM and 11 2) HM x , n 1 x1 1 HM x12,,, x xn HM x AM x , n 1 1 x1 x j1 j 11 1 3) x xx n2 to denote the well known arithmetic, geometric, and 12 n xx12 xn harmonic means, also called the Pythagorean means This follows because PM . The Pythagorean means have the obvious properties: AM x,,, x x 12 n 1 1) PMxx,,, x is independent of order 12 n HM x12,,, x xn 2) PMx,,x , x x 2. Logarithmic and Identric Means 3) PMbxbx12,,, bxnn bPMxx12,,, x The logarithmic mean of two non-negative numbers x 4) PMxx 12, is always a solution of a simple equa- 1 tion. In particular, the arithmetic mean of two numbers and x2 is defined as follows: x1 and x2 can be defined via the equation LM0, x21 LM x ,0 0 AMx12x AM LMxx 11, x 1 The harmonic mean satisfies the same relation with and for positive distinct numbers x and x reciprocals, that is, it is a solution of the equation 1 2 Copyright © 2013 SciRes. APM 332 I. M. IZMIRLI x21 x LM x,, x x LM x, x 012 12 ln x ln x 21 xxxxxx 011220 The following are some basic properties of the loga- 2lx21xxxxxxxxn 0 20 lnln 1 01 2 rithmic means: 1) Logarithmic mean LMab , can be thought of as The identric mean of two distinct positive real num- the mean-value of the function f xx ln over the bers x12, x is defined as: 1 interval ab, . x 1 x 2 x21x 2) The logarithmic mean can also be interpreted as the IM x, x 2 12 x1 area under an exponential curve. e x1 Since with IMxx 11, x 1. 11t The slope of the secant line joining the points 1tt yxy xyddt xt a, f a and bf, b on the graph of the function 00x ln xy ln f xx ln x is the natural logarithm of IM a, b . We also have the identity It can be generalized to more variables according by 1 the mean value theorem for divided differences. LM x,d y x1tt y t 0 3. The Main Theorem Using this representation it is easy to show that Theorem 1. Suppose fa:, b R is a function with LMcxcy,(,) cLMxy a strictly increasing derivative. Then 1) We have the identity b ftd t 22 LM x, y a AMxy, LM x, y ba f bfabfbafa fs s 2 ba ba which follows easily: for all s t in ab, . 22 LM x, y x22yxy Let s be defin ed b y th e equation 0 LMxy, 2 ln xy ln lnxy ln f bfa fs xy 0 ba 2 Then, b To define the logarithmic mean of positive numbers ftdt x01,,xx ,n , we first recall the definition of divided dif- a ferences for a function f x at points x ,,xx , , 01 n ba f bfabfbafa denoted as fs00 s 2 ba ba x01,,xxf ,n is the sharpest form of the above inequality. For 0 mn Proof. By the Mean Value Theorem, for all s,t in a,b , we have xmm ffx ft fs and for 0mn1j and 11j n , f u ts xxfmmj,, for some u between s and t . Assum ing without loss of generality s t, by the assumption of the theorem x ,,xfx ,,x f mmj1mmj1we have xx mj m f tfstsft We now define Integrating both sides with respect to t , we have b LM x,, x 0 n f ttd bafsbfbafa 1 n a n1 1, nx0 ,lnxn b s fb fa ft d t So for example for n = 2, we get a Copyright © 2013 SciRes. APM I. M. IZMIRLI 333 and the inequality of the theorem follows. Thus we have Let us now put ba ba 31 g sbafs bfbafa ab 2 abH 23 ab b s fb fa ftd t implying a Note that H 2 ab gsbafs fbfa or Moreover, since HMab ,, GMab b 1 g bga bafaf b ftt d Let us let ft . The condition of Theorem 1 is a t there exists an s0 in ab, such that gs0 0 . satisfied. We can easily compute the s0 of the theorem Since f is strictly increasing, we have from the equation 11 gsgs 0 0 1 for s s ba 0 2 and s0 ba as gs gs 0 0 for s s0 s0 ab Thus, s is a minimum of g and g sgs 0 0 Our inequality becomes for all sinab , . ba 11 lnba ln ab 4. Applications to Mean Inequalities 2 ab ab We will extend the well-known chain of inequalities Implying, HMab,,, GMa b Aab lnb ln a1 to the more refined ba ab HMab,,, GMab LMab that is IM a,, b A a b GMab ,, LMab using nothing more than the mean value theorem of dif- Now let f tt ln . Again the condition of Theo- ferential calculus. All of these are strict inequalities rem 1 is satisfied. The s0 of the theorem can be com- unless, of course, the numbers are the same, in which puted from the equation case all means are equal to the common value of t he two 1lnlnba numbers. sb a Let us now assume that 0.ab 0 as 1 Let us let ft . The condition of the Theorem 1 s L t 2 0 is satisfied. Solving the equation where LLMab , Since 11 2 22 b ba lntt d b a bln b a ln a s3 ba 0 a we find bb 13 ba ln sabH a 0 a where H HM a,. b Thus, Hence the left-hand side of the inequality becomes b ba 121 lntt d b a ln I a 2 abH23 abH 23 ab where I IM a,. b Copyright © 2013 SciRes. APM 334 I. M. IZMIRLI Consequently our inequality becomes or ba 1 IMab ,, AMab baln I ln LL ln eI 2 L Thus, we now have for ab implying HMab ,,, GMab LMab lnI ln L IM ab,, AM ab that is, LMa,,b IMab REFERENCES Finally, let us put f ttt ln . Again the condition of [1] G. H. Hardy, J. E. Littlewood and G. Pólya, “Inequali- Theorem 1 is satisfied. Since in this case ties,” 2nd Edition, Cambridge University Press, London, 1964. fb fa() bbln aaln 1lnI [2] B. C. Carlson, “A Hypergeometric Mean Value,” Pro- ba ba ceedings of the American Mathematical Society, Vol. 16, No. 4, 1965, pp. 759-766. the s0 of the theorem can be co mputed as doi:10.1090/S0002-9939-1965-0179389-6 s0 I [3] B. C. Carlson and M. D. Tobey, “A Property of the Hy- pergeometric Mean Value,” Proceedings of the American The right-hand side of the inequality becomes Mathematical Society, Vol. 19, No. 2, 1968, pp. 255-262. ba b22ln ba ln a doi:10.1090/S0002-9939-1968-0222349-X I 2 ba [4] E. F. Beckenbach and R. Bellman, “Inequalities,” 3rd Edition, Springer-Verlag, Berlin and New York, 1971. The integral on the left-hand side of our inequality [5] H. Alzer, “Übereinen Wert, der zwischendemgeomet- yields rischen und demartihmetischen Mittelzweier Zahlenliegt,” Elemente der Mathematik, Vol. 40, 1985, pp. 22-24. b 1122 22 tttln d b ln ba ln a b a [6] H. Alzer, “Ungleichungenfür eaab be ,” Elemente a 24 der Mathematik, Vol. 40, 1985, pp. 120-123. implying 1 ba I 2 Copyright © 2013 SciRes. APM .
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