In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, there are two different results that share the common name of the Ky Fan inequality. One is an
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
involving the
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
and
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
of two sets of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s of the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
. The result was published on page 5 of the book ''Inequalities'' by
Edwin F. Beckenbach
Edwin Ford Beckenbach (July 18, 1906 – September 5, 1982) was an American mathematician.
Biography
Beckenbach was born July 18, 1906 in Oak Cliff, Dallas County, Texas, the son of a leather worker and on his father's side the grandson of immi ...
and
Richard E. Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founde ...
(1961), who refer to an unpublished result of
Ky Fan
Ky Fan (樊𰋀, , September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician. He was a professor of mathematics at the University of California, Santa Barbara.
Biography
Fan was born in Hangzhou, the capital of Zhejian ...
. They mention the result in connection with the
inequality of arithmetic and geometric means
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
and
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
's proof of this inequality by forward-backward-induction; a method which can also be used to prove the Ky Fan inequality.
This 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.
The second Ky Fan inequality is used in
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
to investigate the existence of an equilibrium.
Statement of the classical version
If with
for ''i'' = 1, ..., ''n'', then
:
with equality if and only if ''x''
1 = ''x''
2 = · · · = ''x
n''.
Remark
Let
:
denote the arithmetic and geometric mean, respectively, of ''x''
1, . . ., ''x
n'', and let
:
denote the arithmetic and geometric mean, respectively, of 1 − ''x''
1, . . ., 1 − ''x
n''. Then the Ky Fan inequality can be written as
:
which shows the similarity to the
inequality of arithmetic and geometric means
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
given by ''G
n'' ≤ ''A
n''.
Generalization with weights
If ''x
i'' ∈
,½and ''γ
i'' ∈
,1for ''i'' = 1, . . ., ''n'' are real numbers satisfying ''γ''
1 + . . . + ''γ
n'' = 1, then
:
with the convention 0
0 := 0. Equality holds if and only if either
*''γ
ix
i'' = 0 for all ''i'' = 1, . . ., ''n'' or
*all ''x
i'' > 0 and there exists ''x'' ∈ (0,½] such that ''x'' = ''x
i'' for all ''i'' = 1, . . ., ''n'' with ''γ
i'' > 0.
The classical version corresponds to ''γ
i'' = 1/''n'' for all ''i'' = 1, . . ., ''n''.
Proof of the generalization
Idea: Apply
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
to the strictly concave function
:
Detailed proof: (a) If at least one ''x
i'' 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
ix
i'' = 0 for all ''i'' = 1, . . ., ''n''.
(b) Assume now that all ''x
i'' > 0. If there is an ''i'' with ''γ
i'' = 0, then the corresponding ''x
i'' > 0 has no effect on either side of the inequality, hence the ''i''
th term can be omitted. Therefore, we may assume that ''γ
i'' > 0 for all ''i'' in the following. If ''x''
1 = ''x''
2 = . . . = ''x
n'', then equality holds. It remains to show strict inequality if not all ''x
i'' are equal.
The function ''f'' is strictly concave on (0,½], because we have for its second derivative
:
Using the
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
for the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
and Jensen's inequality for the strictly concave ''f'', we obtain that
: