Let $\mathbf{x} = (x_1, \cdots, x_n) \in \mathbb{R}^n$ such that $x_1 \leq \cdots \leq x_n$ and $g: \mathbb{R}^n \to \mathbb{R}$. I'm trying to prove a theorem which requires me to prove that my function $g$ has the following property: $$ g(\mathbf x) \leq g(\sigma \cdot \mathbf x) $$ where $\sigma \cdot \mathbf x$ denotes a permutation of $\mathbf x$. I was wondering if this property of $g$ falls under any particular math term or branch that I can read about online?
My function $g$ has the following format, if it helps: $$g(\mathbf x) = \inf_\mathbf y f(\mathbf x, \mathbf y)$$ where $f(\mathbf x, \cdot)$ is convex.
How about arrangement increasing functions? E.g., http://www.sciencedirect.com/science/article/pii/0047259X88900528