Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, centre $0$. Write $\mathcal{S}$ for the holomorphic injective maps $\{ f : \mathbb{D} \to \mathbb{D} | f(z) = e^{-\lambda} z + O(z^2) \}$ i.e. those that are a (real) scaling near $0$. $\lambda$ is always non-negative and will be called $logcap(f)$ or the logarithmic capacity (from $0$). It is additive, i.e. $logcap(f \circ g) = logcap(f) + logcap(g)$.

I have a sequence $f_n \in \mathcal{S}$, where $f_n$ and $f_{n+1}$ are related as follows:

$f_n = g_1 \circ g_2 \circ \cdots \circ g_k$

$f_{n+1} = h_1 \circ g_1 \circ h_2 \circ g_2 \circ \cdots \circ h_k \circ g_k \circ h_{k+1}$

for some choices of $k$, $g_i$ and $h_i$ (depending on $n$) in $\mathcal{S}$ -- i.e. each $f_i$ is expressed as a sequence of compositions and to get to $f_{i+1}$ we interleave more maps amongst the compositions.

What topologies do we have on $\mathcal{S}$ that give us nice convergence criteria for sequences of the form $f_i$?

If we choose the compact-open topology, which in this case is just the uniform norm, then I can get a long way by assuming that $logcap(f_i)$ is convergent -- i.e. that it is bounded above.

In fact, what I'm specifically stuck on is the following problem

Is $\\|f \circ g - f\|$ bounded uniformly in $f$ in terms of $logcap(g)$?

But if there are any other approaches to the general question I'd be happy to hear them.