Definition:Hölder Mean/Negative Exponent with Zero Parameter

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

Let $p$ be an extended real number.

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

For $p < 0$ and at least one $a_k = 0$, the Hölder mean is defined as:
 * $\ds \map {M_p} {x_1, x_2, \ldots, x_n} = 0$