Papyrus 38

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:

$\prod _{k=1}^n(x_k+y_k{)}^{1/n}\ge \prod _{k=1}^n{x}_k^{1/n}+\prod _{k=1}^n{y}_k^{1/n}$

when x_k, y_k > 0 for all k.

Proof

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

$\prod _{k=1}^n{\left(\frac{x_k}{x_k+y_k}\right)}^{1/n}\le \frac{1}{n}\sum _{k=1}^n\frac{x_k}{x_k+y_k},$

and

$\prod _{k=1}^n{\left(\frac{y_k}{x_k+y_k}\right)}^{1/n}\le \frac{1}{n}\sum _{k=1}^n\frac{y_k}{x_k+y_k}.$

Hence,

$\prod _{k=1}^n{\left(\frac{x_k}{x_k+y_k}\right)}^{1/n}+\prod _{k=1}^n{\left(\frac{y_k}{x_k+y_k}\right)}^{1/n}\le \frac{1}{n}n=1.$

Clearing denominators then gives the desired result.

See also

• Minkowski's inequality

References

• http://eom.springer.de/M/m064060.htm