In mathematics, there are two different results that share the common name of the Ky Fan inequality.
The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who refer to an unpublished result of Ky Fan.
The second Ky Fan inequality is used in game theory to investigate the existence of an equilibrium.
., xn, and let denote the arithmetic and geometric mean, respectively, of 1 − x1, .
Equality holds if and only if either The classical version corresponds to γi = 1/n for all i = 1, .
Equality holds if and only if the right-hand side is also zero, which is the case when γixi = 0 for all i = 1, .
If there is an i with γi = 0, then the corresponding xi > 0 has no effect on either side of the inequality, hence the ith term can be omitted.
It remains to show strict inequality if not all xi are equal.
The function f is strictly concave on (0,1/2], because we have for its second derivative Using the functional equation for the natural logarithm and Jensen's inequality for the strictly concave f, we obtain that where we used in the last step that the γi sum to one.
Taking the exponential of both sides gives the Ky Fan inequality.
This second inequality is equivalent to the Brouwer Fixed Point Theorem, but is often more convenient.
Let S be a compact convex subset of a finite-dimensional vector space V, and let
This Ky Fan Inequality is used to establish the existence of equilibria in various games studied in economics.