Limit of Hölder Mean as Exponent tends to Negative Infinity

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Theorem

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

For $p \in \R_{\ne 0}$, 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:

$\ds \lim_{p \mathop \to -\infty} \map {M_p} {x_1, x_2, \ldots, x_n} = \min \set {x_1, x_2, \ldots, x_n}$


Proof

Let $p \in \R$ such that $p \ne 0$.

Let it be assumed (or arranged) that:

$x_1 \ge x_2 \ge \cdots \ge x_n$

Then:

\(\ds \lim_{p \mathop \to -\infty} \map {M_p} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds \dfrac 1 {\ds \lim_{p \mathop \to +\infty} \map {M_p} {\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 {x_n} } }\)
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\(\ds \) \(=\) \(\ds \dfrac 1 {\paren {\dfrac 1 {x_n} } }\) Limit of Hölder Mean as Exponent tends to Infinity
\(\ds \) \(=\) \(\ds x_n\)
\(\ds \) \(=\) \(\ds \min \set {x_1, x_2, \ldots, x_n}\) by hypothesis: $x_1 \ge x_2 \ge \cdots \ge x_n$

$\blacksquare$


Sources

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