Mahler's Inequality

Theorem

The geometric mean of the termwise sum of two finite sequences of positive numbers is never less than the sum of their two separate geometric means:

$\displaystyle \prod_{k=1}^n \left({x_k + y_k}\right)^{1/n} \ge \prod_{k=1}^n x_k^{1/n} + \prod_{k=1}^n y_k^{1/n}$

where $x_k, y_k > 0$ for all $k$.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \prod_{k=1}^n \left({\frac {x_k} {x_k + y_k} }\right)^{1/n}\)	\(\le\)	\(\displaystyle \frac 1 n \sum_{k=1}^n \frac {x_k} {x_k + y_k}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Arithmetic Mean Never Less than Geometric Mean
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \prod_{k=1}^n \left({\frac {y_k} {x_k + y_k} }\right)^{1/n}\)	\(\le\)	\(\displaystyle \frac 1 n \sum_{k=1}^n \frac {y_k} {x_k + y_k}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Arithmetic Mean Never Less than Geometric Mean
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \prod_{k=1}^n \left({\frac {x_k} {x_k + y_k} }\right)^{1/n} + \prod_{k=1}^n \left({\frac {y_k} {x_k + y_k} }\right)^{1/n}\)	\(\le\)	\(\displaystyle \frac 1 n \sum_{k=1}^n \frac {x_k} {x_k + y_k} + \frac 1 n \sum_{k=1}^n \frac {y_k} {x_k + y_k}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	adding them together
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 n \sum_{k=1}^n \frac {x_k + y_k} {x_k + y_k}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 n n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

This leads to:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \prod_{k=1}^n \left({\frac {x_k} {x_k + y_k} }\right)^{1/n} + \prod_{k=1}^n \left({\frac {y_k} {x_k + y_k} }\right)^{1/n}\)	\(\le\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \frac {\displaystyle \prod_{k=1}^n \left({x_k}\right)^{1/n} } {\displaystyle \prod_{k=1}^n \left({x_k + y_k}\right)^{1/n} } + \frac {\displaystyle \prod_{k=1}^n \left({y_k}\right)^{1/n} } {\displaystyle \prod_{k=1}^n \left({x_k + y_k}\right)^{1/n} }\)	\(\le\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \frac {\displaystyle \prod_{k=1}^n \left({x_k}\right)^{1/n} + \prod_{k=1}^n \left({y_k}\right)^{1/n} } {\displaystyle \prod_{k=1}^n \left({x_k + y_k}\right)^{1/n} }\)	\(\le\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \prod_{k=1}^n \left({x_k}\right)^{1/n} + \prod_{k=1}^n \left({y_k}\right)^{1/n}\)	\(\le\)	\(\displaystyle \prod_{k=1}^n \left({x_k + y_k}\right)^{1/n}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Source of Name

This entry was named for Kurt Mahler.