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Definition D3312
Real power mean

Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in \mathbb{R}$ is a power mean of $x_1, \dots, x_N$ with respect to $p \in \mathbb{R} \setminus \{ 0 \}$ if and only if $$\sum_{n = 1}^N a^p = \sum_{n = 1}^N x^p_n$$

Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in \mathbb{R}$ is a power mean of $x_1, \dots, x_N$ with respect to $p \in \mathbb{R} \setminus \{ 0 \}$ if and only if $$\underbrace{a^p + a^p + \cdots + a^p}_{N \text{ times}} = x^p_1 + x^p_2 + \cdots x^p_N$$
Children
 ▶ Real harmonic mean
Results
 ▶ R5194 ▶ Cauchy-Schwarz inequality for two real sequences ▶ Real geometric mean is a right limit of basic real power means ▶ Weighted Cauchy-Schwarz inequality for two real sequences