{{Short description|Mathematical inequality}} {{For|Ky Fan inequality in game theory|Ky Fan inequality (game theory)}}In mathematics, the term '''Ky Fan inequality''' refers to an inequality involving the geometric mean and arithmetic mean of two sets of real numbers within the unit interval. The result was published on page 5 of the book ''Inequalities'' by Edwin F. Beckenbach and Richard E. Bellman (1961), who attribute it to an unpublished result by Ky Fan. They discuss the inequality in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of that inequality via forward-backward induction—a method that can also be used to prove the Ky Fan inequality.
This inequality is a special case of Levinson's inequality and serves as a foundation for several generalizations and refinements, some of which are referenced below.
==Statement of the classical version== If with <math display=inline> 0\le x_i\le \frac{1}{2} </math> for ''i'' = 1, ..., ''n'', 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''<sub>1</sub> = ''x''<sub>2</sub> = ⋅ ⋅ ⋅ = ''x<sub>n</sub>''.
=== 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''<sub>1</sub>, . . ., ''x<sub>n</sub>'', 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''<sub>1</sub>, . . ., 1 − ''x<sub>n</sub>''. 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 ''G<sub>n</sub>'' ≤ ''A<sub>n</sub>''.
==Generalization with weights== If ''x<sub>i</sub>'' ∈ [0,{{sfrac|1|2}}] 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
:<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 0<sup>0</sup> := 0. Equality holds if and only if either *''γ<sub>i</sub>x<sub>i</sub>'' = 0 for all ''i'' = 1, . . ., ''n'' or *all ''x<sub>i</sub>'' > 0 and there exists ''x'' ∈ (0,{{sfrac|1|2}}] such that ''x'' = ''x<sub>i</sub>'' for all ''i'' = 1, . . ., ''n'' with ''γ<sub>i</sub>'' > 0.
The classical version corresponds to ''γ<sub>i</sub>'' = 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 ''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''.
(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.
The function ''f'' is strictly concave on (0,{{sfrac|1|2}}], 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 ''γ<sub>i</sub>'' sum to one. Taking the exponential of both sides gives the Ky Fan inequality.
==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 = https://link.springer.com/article/10.1007/BF01836094 |mr = 972289 |doi = 10.1007/BF01836094 |s2cid = 122304838 |url-access= subscription }} *{{cite book | last = Beckenbach | first = Edwin Ford |author2=Bellman, Richard Ernest |authorlink2=Richard E. Bellman | title = Inequalities | publisher = Springer-Verlag | year = 1961 | location = Berlin–Göttingen–Heidelberg |mr=158038 | isbn = 978-3-7643-0972-5 }}
*{{cite journal | last = Moslehian | first = M. S. | title = Ky Fan inequalities | journal = Linear and Multilinear Algebra | volume = to appear | arxiv = 1108.1467| bibcode = 2011arXiv1108.1467S | year = 2011 }}
*{{cite journal | last = Neuman | first = Edward |author2=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 |mr=1880271 | doi = 10.7153/mia-05-06 | doi-access = free }}
*{{cite journal | last = Neuman | first = Edward | author2 = 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 | date = August 2005 | url = http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5068-NeSa.pdf | mr = 2162296 | doi = 10.1017/S0004972700034894 | doi-access = free | archive-date = 2020-03-25 | access-date = 2007-06-14 | archive-url = https://web.archive.org/web/20200325213707/http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5068-NeSa.pdf | url-status = dead }} *{{cite journal | last = Sándor | first = József |author2=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 |mr=1717045 | doi = 10.7153/mia-02-43 | doi-access = free }}
==External links== *{{Mathgenealogy|name = Ky Fan|id = 15631}}
Category:Inequalities (mathematics) Category:Articles containing proofs