Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where 
$a,b,c,d \in R$, and $f'(x)>0$ for all $x \in [-1,0]$.


Notice that given a fixed number $c>0$ (and you may suppose that $c \neq 1$ if necessary), there is a unique FLT $F \in \mathcal{S}$ such that the following three conditions hold:


$F(-1)=-1$, $F(0)=0$, and $\dfrac{F'(0)}{F'(-1)}=c^2$


I would like to ask for examples of other classes of maps $\mathcal{C}$ (that is other than the class of FLT's $\mathcal{S}$) such that:


1. $f'(x)>0$ for $x \in [-1,0]$ for all $f \in \mathcal{C}$.
2. The same property as above of there existing a unique element in the class $\mathcal{C}$ satisfying the three conditions above holds.
3. Additionally, the class of maps $\mathcal{C}$ satisfies the property of being closed under (functional) composition of its members.


Thank you in advance to all those who respond,


E(up)lio M.



A Fractional Linear Transformation Class Property
You're asking for a one-parameter family $\mathcal{C}$ of maps on a closed interval $I$ such that 


1. the endpoints of $I$ are fixed by every map in $\mathcal{C}$;
2. the maps in $\mathcal{C}$ are strictly increasing;
3. $\mathcal{C}$ is closed under composition;
4. every positive real number appears exactly once as the ratio of the derivatives of $\mathcal{C}$ at the right and left endpoints of $I$.


The fourth condition means that $\mathcal{C}$ should be parametrised by $\mathbb{R}^+$, so let's write $\mathcal{C} = \{ \psi\_c \mid c>0 \}$, where $c$ represents the square root of the ratio between the derivates at the endpoints, as in your question. The chain rule together with the third condition means that we should require $$\psi\_{c\_1 c\_2} = \psi\_{c\_1} \circ \psi\_{c\_2}$$ for all $c\_1,c\_2>0$. We can reparametrise $\mathcal{C}$ by putting $t=\log c$ (or $t=-\log c$) and writing $\phi\_t = \psi\_c$, this becomes
$$ \phi\_{t\_1+t\_2} = \phi\_{t\_1} \circ \phi\_{t\_2}.$$
In other words, the family $\mathcal{C}$ defines an action of $\mathbb{R}$ on $I$, or in the language of dynamical systems/ODEs, a flow on $I$. This flow comes from integrating a vector field $V(x)$, or if you prefer, solving the ODE 
$$(\*) \qquad \qquad \frac d{dt} \phi\_t(x) = V(\phi\_t(x)).\qquad\qquad\qquad$$
Conditions 1 and 2 mean that $V$ should vanish at the endpoints of $I$ and should be positive everywhere else. But modulo some regularity concerns, that's all you need. If you let $V\colon I\to [0,\infty)$ be any Lipschitz continuous function that vanishes at the endpoints of $I$ and is positive on its interior, then solving $(\*)$ will give you a family of smooth maps (the time-$t$ maps of the flow) that satisfy your conditions.


Incidentally, you can use $(\*)$ to see how both the family $\mathcal{S}$ of FLTs and the family in Robert Israel's (second) answer fit into this scheme. If you write
$$ \phi\_t(x) = \frac{e^{-t} x}{1+(1-e^{-t})x}$$
for the FLT satisfying your conditions (with $c=e^{-t}$ and $t\in\mathbb{R}$ arbitrary), then an easy calculation shows that $\phi\_{s+t} = \phi\_s \circ \phi\_t$, so this defines a flow, and we can compute
$$
V(x) = \frac{d\phi\_t(x)}{dt}|\_{t=0} = -x(1+x).
$$
In other words, these FLTs are just the time-$t$ maps of the logistic flow on the interval $[-1,0]$. You can play a similar game with the arctan maps, but I haven't worked it out to see if the formula comes out cleanly.


Actually, the family of arctan maps in Robert Israel's answer illustrates a (superficially) different way of approaching the question, which is to fix a homeomorphism $h$ between the interior of $I$ and the real line (or the half-line) and then define the family of maps $\mathcal{C}$ to be those that are conjugated to translations (or homotheties) by $h$. So in his example, you conjugate to the negative half-line by $h(x) = \tan(\pi x/2)$ and then let $g\_c = h^{-1} \circ (y\mapsto cy) \circ h$.


Of course, logs turn the half-line into the line and multiplication into addition, so this turns into translation on the line under another change of coordinates. And finding a change of coordinates that turns a map into translation on the line is just another way of saying that you find that map as a solution of an ODE, so it boils down to what I described above. Your choice... you can choose the vector field $V$, or you can choose the homeomorphism $h$. Either will give you quite a large number of examples.


