In mathematics , the Ky-Fan inequality is an inequality that was discovered by Ky Fan and published for the first time by ( Lit .: Beckenbach and Bellman, 1983). Its importance lies primarily in the fact that, due to its similarity to the inequality of the arithmetic and geometric mean, it is the starting point for further generalizations.
In its simplest form, the Ky-Fan inequality is as follows:
If for numbers are with , then applies
.
The equal sign applies if and only if .
If one designates with the arithmetic mean and with the geometric mean of the numbers as well as with the arithmetic mean and with the geometric mean of the numbers , then the Ky-Fan inequality takes the form
on; the similarity to the inequality of the arithmetic and geometric mean becomes clear.
proof
A simple proof of the Ky-Fan inequality is obtained by applying Jensen's inequality to the function that is for concave . This proof immediately provides a generalization of the Ky-Fan inequality with weighted averages:
,
where for the weights and must apply.
Related inequalities
( Ref : Wang and Wang, 1984), the Ky Fan inequality have on the harmonic mean values and extended:
W. Wang and P. Wang: A class of inequalities for the symmetric functions (Chinese) Acta Math Sinica 27 (1984) 485-497, quoted in. ( Ref : Alzer, 1988).