The AM-GM Inequality

Theorem 1 (The AM-GM Inequality): Let $x_1, x_2, ..., x_n$ be nonnegative real numbers. Then $\displaystyle{\frac{x_1 + x_2 + ... + x_n}{n} \geq \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}}$.

"AM-GM" stands for "Arithmetic Mean - Geometric Mean".

• Proof: Consider the function $f(x) = -\ln x$. This function is strictly concave on $(0, \infty)$. By Jensen's Inequality we have that for all $x_1, x_2, ..., x_n \in (0, \infty)$ and for all positive real numbers $t_1, t_2, ..., t_n$ with $t_1 + t_2 + ... + t_n = 1$ we have:

(1)

\begin{align} \quad -\ln (t_1x_1 + t_2x_2 + ... + t_nx_n) \leq -t_1 \ln x_1 - t_2 \ln x_2 - ... - t_n \ln x_n \end{align}

• Or equivalently:

(2)

\begin{align} \quad t_1 \ln x_1 + t_2 \ln x_2 + ... + t_n \ln x_n \leq \ln (t_1x_1 + t_2x_2 + ... + t_nx_n) \end{align}

• Take $t_1 = t_2 = ... = t_n = \frac{1}{n}$. Then:

(3)

\begin{align} \quad \frac{\ln x_1 + \ln x_2 + ... + \ln x_n}{n} \leq \ln \left ( \frac{x_1 + x_2 + ... x_n}{n} \right ) \end{align}

• Using logarithm laws we get:

(4)

\begin{align} \quad \frac{\ln x_1 \cdot x_2 \cdot ... \cdot x_n}{n} \leq \ln \left ( \frac{x_1 + x_2 + ... + x_n}{n} \right ) \end{align}

• And:

(5)

\begin{align} \quad \ln \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n} \leq \ln \left ( \frac{x_1 + x_2 + ... + x_n}{n} \right ) \end{align}

• Therefore:

(6)

\begin{align} \frac{x_1 + x_2 + ... + x_n}{n} \geq \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n} \quad \blacksquare \end{align}