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Mahler's inequality

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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:

∏ k = 1 n ( x k + y k ) 1 / n ≥ ∏ k = 1 n x k 1 / n + ∏ k = 1 n y k 1 / n

when xk, yk > 0 for all k.

Proof

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

∏ k = 1 n ( x k x k + y k ) 1 / n ≤ 1 n ∑ k = 1 n x k x k + y k ,

and

∏ k = 1 n ( y k x k + y k ) 1 / n ≤ 1 n ∑ k = 1 n y k x k + y k .

Hence,

∏ k = 1 n ( x k x k + y k ) 1 / n + ∏ k = 1 n ( y k x k + y k ) 1 / n ≤ 1 n n = 1.

Clearing denominators then gives the desired result.

References

Mahler's inequality Wikipedia
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