Glossary of probability and statistics

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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book ''Inequalities'' by [[Edwin F. Beckenbach|Beckenbach]] and [[Richard E. Bellman|Bellman]] (1961), who refer to an unpublished result of [[Ky Fan]].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.

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.

==Statement of the classical version==
If ''xi'' with 0 ≤ ''xi'' ≤ ½ for ''i'' = 1, ..., ''n'' are real numbers, then

:<math> \frac{ \bigl(\prod_{i=1}^n x_i\bigr)^{1/n} }
{ \bigl(\prod_{i=1}^n (1-x_i)\bigr)^{1/n} }
\le
\frac{ \frac1n \sum_{i=1}^n x_i }
{ \frac1n \sum_{i=1}^n (1-x_i) }
</math>

with equality if and only if ''x''1 = ''x''2 = . . . = ''xn''.

==Remark==
Let
:<math>A_n:=\frac1n\sum_{i=1}^n x_i,\qquad G_n=\biggl(\prod_{i=1}^n x_i\biggr)^{1/n}</math>

denote the arithmetic and geometric mean, respectively, of ''x''1, . . ., ''xn'', and let

:<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>

denote the arithmetic and geometric mean, respectively, of 1 − ''x''1, . . ., 1 − ''xn''. Then the Ky Fan inequality can be written as

:<math>\frac{G_n}{G_n'}\le\frac{A_n}{A_n'},</math>

which shows the similarity to the [[inequality of arithmetic and geometric means]] given by ''Gn'' ≤ ''An''.

==Generalization with weights==
If ''xi'' ∈ [0,½] and ''γi'' ∈ [0,1] for ''i'' = 1, . . ., ''n'' are real numbers satisfying ''γ''1 + . . . + ''γn'' = 1, then

:<math> \frac{ \prod_{i=1}^n x_i^{\gamma_i} }
{ \prod_{i=1}^n (1-x_i)^{\gamma_i} }
\le
\frac{ \sum_{i=1}^n \gamma_i x_i }
{ \sum_{i=1}^n \gamma_i (1-x_i) }
</math>

with the convention 00 := 0. Equality holds if and only if either
*''γixi'' = 0 for all ''i'' = 1, . . ., ''n'' or
*all ''xi'' > 0 and there exists ''x'' ∈ (0,½] such that ''x'' = ''xi'' for all ''i'' = 1, . . ., ''n'' with ''γi'' > 0.

The classical version corresponds to ''γi'' = 1/''n'' for all ''i'' = 1, . . ., ''n''.

==Proof of the generalization==
'''Idea:''' Apply [[Jensen's inequality]] to the strictly concave function

:<math>f(x):= \ln x-\ln(1-x) = \ln\frac x{1-x},\qquad x\in(0,\tfrac12].</math>

'''Detailed proof:''' (a) If at least one ''xi'' 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 ''γixi'' = 0 for all ''i'' = 1, . . ., ''n''.

(b) Assume now that all ''xi'' > 0. If there is an ''i'' with ''γi'' = 0, then the corresponding ''xi'' > 0 has no effect on either side of the inequality, hence the ''i''th term can be omitted. Therefore, we may assume that ''γi'' > 0 for all ''i'' in the following. If ''x''1 = ''x''2 = . . . = ''xn'', then equality holds. It remains to show strict inequality if not all ''xi'' are equal.

The function ''f'' is strictly concave on (0,½], because we have for its second derivative

:<math>f''(x)=-\frac1{x^2}+\frac1{(1-x)^2}<0,\qquad x\in(0,\tfrac12).</math>

Using the [[functional equation]] for the [[natural logarithm]] and Jensen's inequality for the strictly concave ''f'', we obtain that

:<math>
\begin{align}
\ln\frac{ \prod_{i=1}^n x_i^{\gamma_i}}
{ \prod_{i=1}^n (1-x_i)^{\gamma_i} }
&=\ln\prod_{i=1}^n\Bigl(\frac{x_i}{1-x_i}\Bigr)^{\gamma_i}\\
&=\sum_{i=1}^n \gamma_i f(x_i)\\
&<f\biggl(\sum_{i=1}^n \gamma_i x_i\biggr)\\
&=\ln\frac{ \sum_{i=1}^n \gamma_i x_i }
{ \sum_{i=1}^n \gamma_i (1-x_i) },
\end{align}
</math>

where we used in the last step that the ''γi'' sum to one. Taking the exponential of both sides gives the Ky Fan inequality.

==The Ky Fan Inequality in Game Theory==

A second inequality is also called the Ky Fan Inequality, because of a 1972 paper, "A minimax inequality and its applications".
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''.
Then there exists ''x* ∈ S '' such that for all ''y ∈ S, f( x* , y ) ≤ 0 ''. This Ky Fan Inequality is used to establish the existence of
equilibria in various games studied in economics.

==References==
*{{cite journal
| last = Alzer
| first = Horst
| title = Verschärfung einer Ungleichung von Ky Fan
| journal = Aequationes Mathematicae
| volume = 36
| issue = 2-3
| pages = 246–250
| year = 1988
| url = http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D171447
| id = {{MathSciNet | id = 89j:26014}}
| doi = 10.1007/BF01836094
}}

*{{cite book
| last = Beckenbach
| first = Edwin Ford
| coauthors = [[Richard E. Bellman|Bellman, Richard Ernest]]
| title = Inequalities
| publisher = Springer-Verlag
| year = 1961
| location = Berlin–Göttingen–Heidelberg
| id = {{MathSciNet | id = 28:1266}}
| isbn = 3-7643-0972-5
}}

*{{cite journal
| last = Moslehian
| first = M. S.
| title = Ky Fan inequalities
| journal = Linear and Multilinear Algebra
| volume = to appear
| url = http://arxiv.org/abs/1108.1467
}}

*{{cite journal
| last = Neuman
| first = Edward
| coauthors = Sándor, József
| title = On the Ky Fan inequality and related inequalities I
| journal = Mathematical Inequalities & Applications
| volume = 5
| issue = 1
| pages = 49–56
| year = 2002
| url = http://www.ele-math.com/files/mia/05-1/full/mia-05-06.pdf
| id = {{MathSciNet | id = 2002m:26026}}
}}

*{{cite journal
| last = Neuman
| first = Edward
| coauthors = Sándor, József
| title = On the Ky Fan inequality and related inequalities II
| journal = Bulletin of the Australian Mathematical Society
| volume = 72
| issue = 1
| pages = 87–107
| publisher = Australian Mathematical Publishing Assoc. Inc.
|date=August 2005
| url = http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5068-NeSa.pdf
| id = {{MathSciNet | id = 2006d:26031}}
| doi = 10.1017/S0004972700034894
}}

*{{cite journal
| last = Sándor
| first = József
| coauthors = Trif, Tiberiu
| title = A new refinement of the Ky Fan inequality
| journal = Mathematical Inequalities & Applications
| volume = 2
| issue = 4
| pages = 529–533
| year = 1999
| url = http://www.ele-math.com/files/mia/02-4/full/mia-02-43.pdf
| id = {{MathSciNet | id = 2000h:26034}}
}}

==External links==
*{{Mathgenealogy|name = Ky Fan|id = 15631}}

[[Category:Inequalities]]
[[Category:Articles containing proofs]]

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