Inequality of Hölder Means

Theorem
Let $p, q \in \R_{\ne 0}$ be non-zero real numbers with $p < q$.

Let $x_1, x_2, \ldots, x_n \ge 0$ be real numbers.

If $p < 0$, then we require that $x_1, x_2, \ldots, x_n > 0$.

Then the Hölder mean satisfies the inequality:
 * $\map {M_p} {x_1, x_2, \ldots, x_n} \le \map {M_q} {x_1, x_2, \ldots, x_n}$

Equality holds if and only if $x_1 = x_2 = \cdots = x_n$.

Proof
For real $p \ne 0$, the Hölder mean is defined as:
 * $\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n x_k^p}^{1 / p}$

whenever the above expression is defined.

Consider the function $\phi: \R_{\ge 0} \to \R_{\ge 0}$ defined as:
 * $\forall x \in \R_{\ge 0}: \map \phi x = x^{q/p}$

By the Power Rule for Derivatives:
 * $\map {D_x} {x^{q / p} } = \dfrac q p x^{q / p - 1}$

From Real Function is Strictly Convex iff Derivative is Strictly Increasing:
 * $\phi$ is strictly convex if $q > 0$.

From Real Function is Strictly Concave iff Derivative is Strictly Decreasing:
 * $\phi$ is strictly concave if $q < 0$.

Now apply Jensen's inequality to $x_1^p, x_2^p, \ldots, x_n^p$.

For $q > 0$, this gives:
 * $\ds \paren {\frac 1 n \sum_{k \mathop = 1}^n x_k^p}^{q/p} \le \frac 1 n \sum_{k \mathop = 1}^n x_k^q$

For $q < 0$, the reverse inequality holds.

Also by Jensen's inequality, equality holds if and only if $x_1 = x_2 = \cdots = x_n$.

In either case, the result follows.