Ky Fan inequality

In mathematics is called Ky Fan inequality of Ky Fan and discovered a first of (lit.: Beckenbach and Bellman, 1983) referred to published inequality. Their importance lies in the fact that it is by their similarity to the inequality of the arithmetic and geometric means starting point for further generalizations.

Formulation

In the simplest form, the Ky Fan inequality is as follows:

Falls are for numbers, then applies

The equality holds if and only if.

If we denote by the arithmetic mean and the geometric mean of the numbers as well as with the arithmetic mean and the geometric mean of the numbers, so does the Ky Fan inequality in the form

To; the similarity to the inequality of the arithmetic and geometric means is thus significantly.

Evidence

A simple proof of the Ky Fan inequality results when applying the Jensen's inequality to the function that is concave. This evidence provides a direct generalization of the Ky Fan inequality with weighted averages:

Which must for the weights and apply.

Related inequalities

( Ref: Wang and Wang, 1984) have the Ky Fan inequality extended to the harmonic mean values ​​and: