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.