I'm trying to prove the inequality between generalised means $$ \left( \frac{1}{n} \sum_{k = 1}^{n} a_k^p \right)^{1/p} \ge \left( \frac{1}{n} \sum_{k = 1}^{n} a_k^q \right)^{1/q} $$ for $p \ge q$ and arbitrary $a_1, \ldots, a_n > 0$. I know it can be done with Jensen's inequality and I understand the reasoning in the proof, but I don't know how to do it using basic methods. I realised that I don't have the right intuition behind this inequality that doesn't rely on the function $\lambda x \in \mathbb{R}_{+}. \; x^{p/q}$ being convex.
Do you happen to have some other intuition for this problem that's more basic?
