What is the condition $|f(x)| \le f(|x|)$ called?

Question

Suppose I have a function $f: \mathbb R \to \mathbb R$ so that for all $x\in \mathbb R$ it holds that $|f(x)| \le f(|x|).$

Does this condition have a name?

Answer 1

(Too long for a comment.)

• For $\,x \ge 0\,$ we have $\,|x| = x\,$ so the condition writes as $\,|f(x)| \le f(x)\,$. Given that $\,|y| \ge y\,$ for any real $\,y\,$ with equality iff $\,y \ge 0\,$, the latter is equivalent to $\,f(x) \ge 0\,$ for all $\,\forall x \ge 0\,$.

• For $\,x \lt 0\,$ the condition is $\,|f(x)| \le f(-x)\,$.

Piecing the two cases together, $\,f\,$ is a function that must be non-negative on $\,\mathbb{R}^+ \cup \{0\}\,$, and whose absolute value is bounded above by $\,f(-x)\,$ on $\,\mathbb{R}^-$. I don't know that such functions have their own dedicated name.

From a different angle, the condition can be written as $\,(g \circ f)(x) \le (f \circ g)(x)\,$ with $\,g(x) = |x|\,$, but a quick search did not find any terminology of "$f$ is over/super/sub/under-commuting with $g$" being used to describe such a relation.