The AM-GM inequality is a known one, it states that for any list of n numbers, the following is true:
$$ \frac{x_1 + x_2 + \text{...} + x_n}{n} \ge \sqrt[n]{x_1 \cdot x_2 \cdot \text{...} \cdot x_n }$$
But there are other important statistic means in math, such as the harmonic mean:
$$ H =\left(\frac{\sum_{k=1}^n x_k^{-1}}{n}\right)^{-1}$$
Or the root mean square:
$$ X_\text{rms} = \sqrt{\frac{1}{n} (x_1^2 + x_2 ^ 2 + ... + x_n^2)} $$
The AM-GM can be extended with these means. More specifically, given any list of numbers:
let A be the arithmetic mean, GM be the geometric mean
let H be the harmonic mean and $X_\text{rms}$ bet the root mean square
The following is true:
$$ X_\text{rms} \ge A \ge GM \ge H $$
All of these inequalities can be proven using mathematical induction, and it is a well-known proof. But I want to consider a different approach:
$ \text{For any list of n numbers, let's define the function }$
$$ S(p) = \left( \frac{x_1^p + x_2^p + \text{...} + x_n^p}{n} \right) ^ \frac{1}{p} $$
$ \text{Then} $ $ H = S(-1) $, $ A = S(1) $, $ X_\text{rms} = S(2)$
$ \text{Notice that}$ $GM = \lim_{p \to 0} S(p) $
So we can say for sure that $S(2) \ge S(1) \ge S(p \to 0) \ge S(-1) $. We then make the hypothesis that $ S(x) \text{is monotonically increasing on the set of real numbers }$
My question is how to prove this hypothesis? Ideally, your answer would also contain a proof of $\lim_{p \to 0} S(p) = GM$ as I figured this one out on pure intuition.
P.S I am in the last grade of school right now, so no high-level math (Higher than basics of calculus or complex numbers theory) would be prefered. Still, though, any help would be very much appreciated!
