Log-nonexpansive functions: terminology and references - MathOverflow

Log-nonexpansive functions: terminology and references

2012-09-12T20:49:13Z

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

(Defn.). Let $h: (0,\infty) \to (0,\infty)$ be strictly positive, continuous function. We say $h$ is log-nonexpansive if \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.

Answer by Dirk for Log-nonexpansive functions: terminology and references

2012-09-13T06:57:46Z

To expand Robert Israel's comment: The class of log-nonexpansinve functions on $]0,\infty[$ is the image of the nonexpansive functions of $\mathbb{R}$ under conjugation with $\exp:\mathbb{R}\to ]0,\infty[$, that is, every log-nonexpansive function $h:]0,\infty[\to]0,\infty[$ is obtained by a nonexpansive function $g$ via $h = \exp\circ g\circ \exp^{-1}$.

Since nonexpansive functions are central in optimization of real-valued functions, it seems natural, that an important class of functions for optimization of positive valued functions is obtained by conjugation with a bijection from $\mathbb{R}$ to $]0,\infty[$.

I wonder, if other bijections from $\mathbb{R}$ to $]0,\infty[$ would also work in your context (although $\exp$ seems very well suited).