Mahler's inequality: Difference between revisions
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In [[mathematics]], '''Mahler's inequality''' states that |
In [[mathematics]], '''Mahler's inequality''' states that |
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:<math>\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}</math> |
:<math>\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}</math> |
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In mathematics, Mahler's inequality states that
when for all .
Proof
By the inequality of arithmetic and geometric means, we have:
- , as well as .
Hence,
- .