Mahler's inequality


In mathematics, Mahler's inequality, named after Kurt Mahler, states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means:
when xk, yk > 0 for all k.

Proof

By the inequality of arithmetic and geometric means, we have:
and
Hence,
Clearing denominators then gives the desired result.