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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.

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If you write $g(x) = \log h(e^x)$, so $g$ is a continuous function from $\mathbb R$ to itself, your condition is equivalent to $g$ being non-expansive in the usual metric on $\mathbb R$. –  Robert Israel Sep 13 '12 at 0:01
    
@Robert: True! I had thought of doing a variable change, but ultimately, I am searching for sufficient conditions that allow one to easily test nonexpansivity for at least a class of functions. And a variable change merely pushes the burden of testing into a different metric.... :-) –  Suvrit Sep 13 '12 at 7:57
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$g$ is non-expansive (in the usual metric on $\mathbb R$) iff $g(x) = g(0) + \int_0^x u(t)\ dt$ where $u$ is a measurable function with $|u| \le 1$ almost everywhere. –  Robert Israel Sep 14 '12 at 3:10
    
Thanks Robert for this result; I did not know this theorem, but it seems that I might be able to use it. :-) –  Suvrit Sep 14 '12 at 7:45
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1 Answer 1

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).

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I guess, then I cannot hope for a more general set. I was looking for a power-series characterization, but I guess, in light of the above conjugation, I need to add more restrictions to allow easy to test "sufficient" conditions of nonexpansivity for my problem. Because with this change of variables, the burden of testing log-nonexpansivity just shifts to testing nonexpansivity, something that I had been trying to avoid. –  Suvrit Sep 13 '12 at 7:56
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