Definition:Hölder Mean/Negative Exponent with Zero Parameter
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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:
- $\ds \map {M_p} {x_1, x_2, \ldots, x_n} = 0$
Also see
- Results about the Hölder mean can be found here.
Source of Name
This entry was named for Otto Ludwig Hölder.
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Generalized Mean: $3.1.15$