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
Here, we prove the case where $0 < p < q$. The other cases can be handled similarly.

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

By the power rule and Derivative of Convex or Concave Function, $\phi$ is strictly convex.

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

This gives:
 * $\displaystyle \left({ \frac 1 n \sum_{k=1}^n x_k^p }\right)^{q/p} \le \frac 1 n \sum_{k=1}^n x_k^q$