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Result R5182 on D2455: Real geometric mean
Subresult of R3568: Real AM-GM inequality

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)	$\begin{equation} \prod_{n = 1}^N x_n = \left( \frac{1}{N} \sum_{n = 1}^N x_n \right)^N \quad \iff \quad x_1 = x_2 = \cdots = x_N \end{equation}$

Subresults

▶	R5211: Tight upper bound to a product of three unsigned real numbers
▶	R5210: Tight upper bound to a product of two unsigned real numbers
▶	R5183: Unique global maximizer for a finite product of unsigned real numbers with a given sum

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.

$\square$