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:
 * $M_p \left({x_1, x_2, \ldots, x_n}\right) \le M_q \left({x_1, x_2, \ldots, x_n}\right)$

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:
 * $\displaystyle M_p \left({x_1, x_2, \ldots, x_n}\right) = \left({\frac 1 n \sum_{k \mathop = 1}^n x_k^p}\right)^{1/p}$

whenever the above expression is defined.

Consider the function:
 * $\phi : \R_{\ge 0} \to \R_{\ge 0}$ defined by $\phi \left({x}\right) = x^{q/p}$

It follows directly from the power rule and Real Function is Strictly Convex iff Derivative is Strictly Increasing that $\phi$ is strictly convex if $q > 0$

Similarly, from Real Function is Strictly Concave iff Derivative is Strictly Decreasing and 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:
 * $\displaystyle \left({ \frac 1 n \sum_{k \mathop = 1}^n x_k^p }\right)^{q/p} \le \frac 1 n \sum_{k \mathop = 1}^n x_k^q$

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.