During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.

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Defn.). Let $h: (0,\infty) \to (0,\infty)$ be strictly positive, continuous function. We say $h$ islog-nonexpansiveif \begin{equation*} |\log h(s)-\log h(t)| \le |\log s - \log t|,\qquad\text{for all}\ \ s, t > 0. \end{equation*}

This definition is just an alternate way of saying that $h$ is non-expansive under the *hyperbolic distance* $d(x,y) := |\log x - \log y|$.

My question is whether there exists a classification of such log-nonexpansive functions, or at least some sufficient conditions that ensure this non-expansivity?

I will also be grateful for references to material where such log-nonexpansivity arises.