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Definition D2455
Real geometric mean

Let $x_1, \dots, x_N \in \mathbb{R}$ each be a D993: Real number.
A D993: Real number $a \in \mathbb{R}$ is a geometric mean of $x_1, \dots, x_N$ if and only if $$\underbrace{a a \cdots a}_{N \text{ times}} = x_1 x_2 \cdots x_N$$

Let $x_1, \dots, x_N \in \mathbb{R}$ each be a D993: Real number.
A D993: Real number $a \in \mathbb{R}$ is a geometric mean of $x_1, \dots, x_N$ if and only if $$\prod_{n = 1}^N a = \prod_{n = 1}^N x_n$$
Results
 ▶ Real AM-GM inequality ▶ Real GM-HM inequality ▶ Real arithmetic expression for unsigned real geometric mean ▶ Tight lower bound to a finite product of positive real numbers ▶ Tight upper bound to a finite product of unsigned real numbers ▶ Tight upper bound to a product of three unsigned real numbers ▶ Tight upper bound to a product of two unsigned real numbers ▶ Weighted real AM-GM inequality ▶ Weighted real GM-HM inequality