ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5182 on D2455: Real geometric mean
Tight upper bound to a finite product of unsigned real numbers
Formulation 0
Let $x_1, \ldots, x_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Then
(1) \begin{equation} \prod_{n = 1}^N x_n \leq \left( \frac{1}{N} \sum_{n = 1}^N x_n \right)^N \end{equation}
(2) \begin{equation} \prod_{n = 1}^N x_n = \left( \frac{1}{N} \sum_{n = 1}^N x_n \right)^N \quad \iff \quad x_1 = x_2 = \cdots = x_N \end{equation}
Proofs
Proof 0
Let $x_1, \ldots, x_N \in [0, \infty)$ each be an D4767: Unsigned real number.
This result is a particular case of R3568: Real AM-GM inequality. $\square$