Mahler's Inequality
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Theorem
The geometric mean of the termwise sum of two finite sequences of positive real numbers is never less than the sum of their two separate geometric means:
- $\ds \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} \ge \prod_{k \mathop = 1}^n x_k^{1/n} + \prod_{k \mathop = 1}^n y_k^{1/n}$
where $x_k, y_k > 0$ for all $k$.
Proof
\(\ds \prod_{k \mathop = 1}^n \paren {\frac {x_k} {x_k + y_k} }^{1/n}\) | \(\le\) | \(\ds \frac 1 n \sum_{k \mathop = 1}^n \frac {x_k} {x_k + y_k}\) | Cauchy's Mean Theorem | |||||||||||
\(\ds \prod_{k \mathop = 1}^n \paren {\frac {y_k} {x_k + y_k} }^{1/n}\) | \(\le\) | \(\ds \frac 1 n \sum_{k \mathop = 1}^n \frac {y_k} {x_k + y_k}\) | Cauchy's Mean Theorem | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \prod_{k \mathop = 1}^n \paren {\frac {x_k} {x_k + y_k} }^{1/n} + \prod_{k \mathop = 1}^n \paren {\frac {y_k} {x_k + y_k} }^{1/n}\) | \(\le\) | \(\ds \frac 1 n \sum_{k \mathop = 1}^n \frac {x_k} {x_k + y_k} + \frac 1 n \sum_{k \mathop = 1}^n \frac {y_k} {x_k + y_k}\) | adding them together | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n \sum_{k \mathop = 1}^n \frac {x_k + y_k} {x_k + y_k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
This leads to:
\(\ds \prod_{k \mathop = 1}^n \paren {\frac {x_k} {x_k + y_k} }^{1/n} + \prod_{k \mathop = 1}^n \paren {\frac {y_k} {x_k + y_k} }^{1/n}\) | \(\le\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\ds \prod_{k \mathop = 1}^n \paren {x_k}^{1/n} } {\ds \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} } + \frac {\ds \prod_{k \mathop = 1}^n \paren {y_k}^{1/n} } {\ds \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} }\) | \(\le\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\ds \prod_{k \mathop = 1}^n \paren {x_k}^{1/n} + \prod_{k \mathop = 1}^n \paren {y_k}^{1/n} } {\ds \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} }\) | \(\le\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \prod_{k \mathop = 1}^n \paren {x_k}^{1/n} + \prod_{k \mathop = 1}^n \paren {y_k}^{1/n}\) | \(\le\) | \(\ds \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n}\) |
$\blacksquare$
Source of Name
This entry was named for Kurt Mahler.