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| In [[mathematics]], there are two different results that share the common name of the '''Ky Fan inequality'''. One is an [[inequality (mathematics)|inequality]] involving the [[geometric mean]] and [[arithmetic mean]] of two sets of [[real number]]s of the [[unit interval]]. The result was published on page 5 of the book ''Inequalities'' by [[Edwin F. Beckenbach|Beckenbach]] and [[Richard E. Bellman|Bellman]] (1961), who refer to an unpublished result of [[Ky Fan]]. They mention the result in connection with the [[inequality of arithmetic and geometric means]] and [[Augustin Louis Cauchy]]'s proof of this inequality by forward-backward-induction; a method which can also be used to prove the Ky Fan inequality.
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| The Ky Fan inequality is a special case of [[Levinson's inequality]] and also the starting point for several generalizations and refinements, some of them are given in the references below.
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| ==Statement of the classical version==
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| If ''x<sub>i</sub>'' with 0 ≤ ''x<sub>i</sub>'' ≤ ½ for ''i'' = 1, ..., ''n'' are real numbers, then
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| :<math> \frac{ \bigl(\prod_{i=1}^n x_i\bigr)^{1/n} }
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| { \bigl(\prod_{i=1}^n (1-x_i)\bigr)^{1/n} }
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| \le
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| \frac{ \frac1n \sum_{i=1}^n x_i }
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| { \frac1n \sum_{i=1}^n (1-x_i) }
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| </math>
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| with equality if and only if ''x''<sub>1</sub> = ''x''<sub>2</sub> = . . . = ''x<sub>n</sub>''.
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| ==Remark==
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| Let
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| :<math>A_n:=\frac1n\sum_{i=1}^n x_i,\qquad G_n=\biggl(\prod_{i=1}^n x_i\biggr)^{1/n}</math>
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| denote the arithmetic and geometric mean, respectively, of ''x''<sub>1</sub>, . . ., ''x<sub>n</sub>'', and let
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| :<math>A_n':=\frac1n\sum_{i=1}^n (1-x_i),\qquad G_n'=\biggl(\prod_{i=1}^n (1-x_i)\biggr)^{1/n}</math>
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| denote the arithmetic and geometric mean, respectively, of 1 − ''x''<sub>1</sub>, . . ., 1 − ''x<sub>n</sub>''. Then the Ky Fan inequality can be written as
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| :<math>\frac{G_n}{G_n'}\le\frac{A_n}{A_n'},</math>
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| which shows the similarity to the [[inequality of arithmetic and geometric means]] given by ''G<sub>n</sub>'' ≤ ''A<sub>n</sub>''.
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| ==Generalization with weights==
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| If ''x<sub>i</sub>'' ∈ [0,½] and ''γ<sub>i</sub>'' ∈ [0,1] for ''i'' = 1, . . ., ''n'' are real numbers satisfying ''γ''<sub>1</sub> + . . . + ''γ<sub>n</sub>'' = 1, then
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| :<math> \frac{ \prod_{i=1}^n x_i^{\gamma_i} }
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| { \prod_{i=1}^n (1-x_i)^{\gamma_i} }
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| \le
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| \frac{ \sum_{i=1}^n \gamma_i x_i }
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| { \sum_{i=1}^n \gamma_i (1-x_i) }
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| </math>
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| with the convention 0<sup>0</sup> := 0. Equality holds if and only if either
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| *''γ<sub>i</sub>x<sub>i</sub>'' = 0 for all ''i'' = 1, . . ., ''n'' or
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| *all ''x<sub>i</sub>'' > 0 and there exists ''x'' ∈ (0,½] such that ''x'' = ''x<sub>i</sub>'' for all ''i'' = 1, . . ., ''n'' with ''γ<sub>i</sub>'' > 0.
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| The classical version corresponds to ''γ<sub>i</sub>'' = 1/''n'' for all ''i'' = 1, . . ., ''n''.
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| ==Proof of the generalization==
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| '''Idea:''' Apply [[Jensen's inequality]] to the strictly concave function
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| :<math>f(x):= \ln x-\ln(1-x) = \ln\frac x{1-x},\qquad x\in(0,\tfrac12].</math>
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| '''Detailed proof:''' (a) If at least one ''x<sub>i</sub>'' is zero, then the left-hand side of the Ky Fan inequality is zero and the inequality is proved. Equality holds if and only if the right-hand side is also zero, which is the case when ''γ<sub>i</sub>x<sub>i</sub>'' = 0 for all ''i'' = 1, . . ., ''n''.
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| (b) Assume now that all ''x<sub>i</sub>'' > 0. If there is an ''i'' with ''γ<sub>i</sub>'' = 0, then the corresponding ''x<sub>i</sub>'' > 0 has no effect on either side of the inequality, hence the ''i''<sup>th</sup> term can be omitted. Therefore, we may assume that ''γ<sub>i</sub>'' > 0 for all ''i'' in the following. If ''x''<sub>1</sub> = ''x''<sub>2</sub> = . . . = ''x<sub>n</sub>'', then equality holds. It remains to show strict inequality if not all ''x<sub>i</sub>'' are equal.
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| The function ''f'' is strictly concave on (0,½], because we have for its second derivative
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| :<math>f''(x)=-\frac1{x^2}+\frac1{(1-x)^2}<0,\qquad x\in(0,\tfrac12).</math>
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| Using the [[functional equation]] for the [[natural logarithm]] and Jensen's inequality for the strictly concave ''f'', we obtain that
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| :<math>
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| \begin{align}
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| \ln\frac{ \prod_{i=1}^n x_i^{\gamma_i}}
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| { \prod_{i=1}^n (1-x_i)^{\gamma_i} }
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| &=\ln\prod_{i=1}^n\Bigl(\frac{x_i}{1-x_i}\Bigr)^{\gamma_i}\\
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| &=\sum_{i=1}^n \gamma_i f(x_i)\\
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| &<f\biggl(\sum_{i=1}^n \gamma_i x_i\biggr)\\
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| &=\ln\frac{ \sum_{i=1}^n \gamma_i x_i }
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| { \sum_{i=1}^n \gamma_i (1-x_i) },
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| \end{align}
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| </math>
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| where we used in the last step that the ''γ<sub>i</sub>'' sum to one. Taking the exponential of both sides gives the Ky Fan inequality.
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| ==The Ky Fan Inequality in Game Theory==
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| A second inequality is also called the Ky Fan Inequality, because of a 1972 paper, "A minimax inequality and its applications".
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| This second inequality is equivalent to the [[Brouwer Fixed Point Theorem]], but is often more convenient. Let ''S'' be a [[compact space|compact]] [[convex set|convex]] subset of a finite dimensional [[vector space]] ''V'', and let ''f(x,y)'' be a continuous function from ''S × S'' to the [[real numbers]] that is [[lower semicontinuous]] in ''x'', [[concave function|concave]] in ''y'' and has ''f(z,z) ≤ 0'' for all ''z'' in ''S''.
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| Then there exists ''x<sup>*</sup> ∈ S '' such that for all ''y ∈ S, f( x<sup>*</sup> , y ) ≤ 0 ''. This Ky Fan Inequality is used to establish the existence of
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| equilibria in various games studied in economics.
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| ==References==
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| *{{cite journal
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| | last = Alzer
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| | first = Horst
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| | title = Verschärfung einer Ungleichung von Ky Fan
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| | journal = Aequationes Mathematicae
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| | volume = 36
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| | issue = 2-3
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| | pages = 246–250
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| | year = 1988
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| | url = http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D171447
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| | id = {{MathSciNet | id = 89j:26014}}
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| | doi = 10.1007/BF01836094
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| }}
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|
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| *{{cite book
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| | last = Beckenbach
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| | first = Edwin Ford
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| | coauthors = [[Richard E. Bellman|Bellman, Richard Ernest]]
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| | title = Inequalities
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| | publisher = Springer-Verlag
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| | year = 1961
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| | location = Berlin–Göttingen–Heidelberg
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| | id = {{MathSciNet | id = 28:1266}}
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| | isbn = 3-7643-0972-5
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| }}
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| *{{cite journal
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| | last = Moslehian
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| | first = M. S.
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| | title = Ky Fan inequalities
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| | journal = Linear and Multilinear Algebra
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| | volume = to appear
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| | url = http://arxiv.org/abs/1108.1467
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| }}
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| *{{cite journal
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| | last = Neuman
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| | first = Edward
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| | coauthors = Sándor, József
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| | title = On the Ky Fan inequality and related inequalities I
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| | journal = Mathematical Inequalities & Applications
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| | volume = 5
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| | issue = 1
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| | pages = 49–56
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| | year = 2002
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| | url = http://www.ele-math.com/files/mia/05-1/full/mia-05-06.pdf
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| | id = {{MathSciNet | id = 2002m:26026}}
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| }}
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| *{{cite journal
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| | last = Neuman
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| | first = Edward
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| | coauthors = Sándor, József
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| | title = On the Ky Fan inequality and related inequalities II
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| | journal = Bulletin of the Australian Mathematical Society
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| | volume = 72
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| | issue = 1
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| | pages = 87–107
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| | publisher = Australian Mathematical Publishing Assoc. Inc.
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| |date=August 2005
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| | url = http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5068-NeSa.pdf
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| | id = {{MathSciNet | id = 2006d:26031}}
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| | doi = 10.1017/S0004972700034894
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| }}
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|
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| *{{cite journal
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| | last = Sándor
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| | first = József
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| | coauthors = Trif, Tiberiu
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| | title = A new refinement of the Ky Fan inequality
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| | journal = Mathematical Inequalities & Applications
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| | volume = 2
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| | issue = 4
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| | pages = 529–533
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| | year = 1999
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| | url = http://www.ele-math.com/files/mia/02-4/full/mia-02-43.pdf
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| | id = {{MathSciNet | id = 2000h:26034}}
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| }}
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| ==External links==
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| *{{Mathgenealogy|name = Ky Fan|id = 15631}}
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| [[Category:Inequalities]]
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| [[Category:Articles containing proofs]]
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