Ky Fan inequality

In mathematics, the term Ky Fan inequality refers to an inequality involving the geometric mean and arithmetic mean of two sets of real numbers within the unit interval. The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who attribute it to an unpublished result by Ky Fan. They discuss the inequality in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of that inequality via forward-backward induction—a method that can also be used to prove the Ky Fan inequality.

This inequality is a special case of Levinson's inequality and serves as a foundation for several generalizations and refinements, some of which are referenced below.

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