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:

:$\prod_^n (x_k + y_k)^ \ge \prod_^n x_k^ + \prod_^n y_k^$

when ''x''_''k'', ''y''_''k'' > 0 for all ''k''.

Proof

By the inequality of arithmetic and geometric means, we have:

:$\prod_^n \left(\right)^ \le \sum_^n ,$

and

: $\prod_^n \left(\right)^ \le \sum_^n .$

Hence,

:$\prod_^n \left(\right)^ + \prod_^n \left(\right)^ \le n = 1.$

Clearing denominators then gives the desired result.

See also

*Minkowski inequality

References

Minkowski inequality
in the ''Encyclopedia of Mathematics''

Inequalities
Articles containing proofs

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