An Elementary Proof of the Mean Inequalities

Home , Logarithmic mean, Pythagorean means

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)

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 harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric 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 j1 12 n  x GM n 1  n GM x2 1 n GM x12,,, x xnj GM x x j1 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 j1 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) PMbxbx12,,, bxnn bPMxx12,,, 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 LM0, x21 LM x ,0 0

AMx12x 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

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- 2lx21xxxxxxxxn 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 x21x 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 1tt yxy xyddt xt a, f a and bf,  b on the graph of the function 00x ln xy ln f xx ln x is the natur al 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   x1tt 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 ftd 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 x22yxy 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  ftdt 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  fs00 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 0mn1j and 11j 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  mmj1mmj1we 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 n1  1, nx0 ,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  ftd t implying a Note that H 2  ab gsbafs    fbfa  or Moreover, since HMab ,,  GMab  b 1 g bga  bafaf    b ftt d Let us let ft . The condition of Theorem 1 is a t there exists an s0 in ab,  such that gs0  0 . satisfied. We can easily compute the s0 of the theorem Since f  is strictly increasing, we have from the equation  11 gsgs 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 min imum of g and g sgs 0 0   Our inequality becomes for all sinab , . 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 1lnlnba 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

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