The Arithmetic Mean - Geometric Mean Inequality

There are many problems in engineering and science where we work with the arithmetic mean

$x_{AM} = \frac{x_1 + x_2 + \dots + x_n}{n}$

or the geometric mean

$x_{GM} = \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n}$

While these are two different ways of combining a number of sample points, we can prove the Arithmetic Mean - Geometric Mean (AM-GM) Inequality, which is a useful result that tells us that the arithmetic mean cannot be smaller than the geometric mean, and that these two measures are equal if, and only iff, the numbers are all equal.

Formally, the AM-GM inequality is stated as

For any set of nonnegative real numbers $\{x_1, x_2, \dots, x_n\}$ the arithmetic mean $x_{AM}$ and geometric mean $x_{GM}$ are related by $x_{AM} \geq x_{GM}$ The equality $x_{AM} = x_{GM} = \mu$ holds iff $\forall i \in \{1, 2,\dots,n\}: x_i = \mu$

If we consider the case that $n = 2$ , the AM-GM inequality becomes

For any $x_1, x_2 \geq$ $\frac12\left(x_1 + x_2\right) \geq \sqrt{x_1 x_2}$ The equality holds if and only if $x_1 = x_2$ .

The AM-GM inequality is a useful tool in a number of optimisation problems.

For example, consider an optimsation problem where we are trying to find the values of $x_1$ and $x_2$ so that their product $x_1 x_2$ is maximised.

Suppose we are told the sum of these values can be no greater than some value $S$ . This tells us that

\begin{aligned} x_1 + x_2 &\leq S \\ \frac12\left(x_1 + x_2\right) &\leq \frac{S}2 \\ x_{AM} &\leq \frac{S}2 \end{aligned}

By using the AM-GM inequality $x_{AM} \geq x_{GM}$ , we can now establish that

\begin{aligned} x_{GM} &\leq \frac{S}2 \\ \sqrt{x_1 x_2} &\leq \frac{S}2 \\ x_1 x_2 &\leq \frac{S^2}4 \\ \end{aligned}

The AM-GM inequality doesn't just set the upper bound, it also tells us the conditions under which this maximum will be achieved, which is when $x_1 = x_2$ . Since we know that $x_1 + x_2 = S$ , we can conclude that $x_1 + x_2 = \frac{S}2$ . will give us the largest value of $x_1 x_2$ .

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