Inequality relying on integer partitions and dominance ordering

Question

Let $\lambda$, $\mu$ be two partitions of a natural number $n$, such that $\lambda$ dominates $\mu$ in the usual dominance order on partitions.

I would like to prove that if $q\geq 2$ is a natural number, then $$\sum_{j\in \mu} q^j \leq \sum_{i\in \lambda} q^i $$

I'm sure this must be well-known if it is true, but I cannot find it in the literature. Even a reference for this (or indeed a counterexample!) would be most welcome.

Answer 1

This is an immediate consequence of Karamata's inequality since $x \mapsto q^x$ is convex for $q \geq 1$. (Credit where credit is due: I essentially learned this from Fedor Petrov.)