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 as many examples as possible of other classes of maps $\mathcal{C}$ (that is other than the class of FLT's) such that:
$F'(x)>0$ for $x \in [-1,0]$.
The same property as above of there existing a unique element in the class $\mathcal{C}$ satisfying the three conditions above holds.
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.