Minimum is Less than or Equal to Hölder Mean

Theorem
Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be positive real numbers.

Let $p \in \R$ be a real number.

Let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.

Then:
 * $\min \set {x_1, x_2, \ldots, x_n} \le \map {M_p} {x_1, x_2, \ldots, x_n}$

Equality holds :
 * $x_1 = x_2 = \cdots x_n$

or:
 * $p < 0$ and $x_k = 0$ for some $k \in \set {1, 2, \ldots, n}$.