Ky Fan inequality

Ky Fan inequality

In mathematics, the Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book "Inequalities" by Beckenbach and Bellman (1961), who refer to an unpublished result of Ky Fan.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.

tatement of the classical version

If "xi" with 0 ≤ "xi" ≤ ½ for "i" = 1, ..., "n" are real numbers, then

: frac{ igl(prod_{i=1}^n x_iigr)^{1/n} } { igl(prod_{i=1}^n (1-x_i)igr)^{1/n} } le frac{ frac1n sum_{i=1}^n x_i } { frac1n sum_{i=1}^n (1-x_i) }

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

Remark

Let:A_n:=frac1nsum_{i=1}^n x_i,qquad G_n=iggl(prod_{i=1}^n x_iiggr)^{1/n}

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

:A_n':=frac1nsum_{i=1}^n (1-x_i),qquad G_n'=iggl(prod_{i=1}^n (1-x_i)iggr)^{1/n}

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

:frac{G_n}{G_n'}lefrac{A_n}{A_n'},

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

: 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) }

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

:f(x):= ln x-ln(1-x) = lnfrac x{1-x},qquad xin(0, frac12] .

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

:f"(x)=-frac1{x^2}+frac1{(1-x)^2}<0,qquad xin(0, frac12).

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

:egin{align}lnfrac{ prod_{i=1}^n x_i^{gamma_i { prod_{i=1}^n (1-x_i)^{gamma_i} }&=lnprod_{i=1}^nBigl(frac{x_i}{1-x_i}Bigr)^{gamma_i}\&=sum_{i=1}^n gamma_i f(x_i)\&

where we used in the last step that the "γi" 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
pages = 246–250
year = 1988
url = http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D171447
id = MathSciNet | id = 89j:26014
*cite book
last = Beckenbach
first = Edwin Ford
coauthors = Bellman, Richard Ernest
title = Inequalities
publisher = Springer-Verlag
date = 1961
location = Berlin–Göttingen–Heidelberg
id = MathSciNet | id = 28:1266

*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.
month = August
year = 2005
url = http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5068-NeSa.pdf
id = MathSciNet | id = 2006d:26031
*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


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