Let $G \subseteq \mathbb{C}$ be a bounded domain in $\mathbb{C}$. Consider for $m \in \mathbb{N}$ a sequence of quasiconformal mappings $f_m: G \longrightarrow \mathbb{C}$ with unbounded maximal dilatation, i.e. $K(f_m) \longrightarrow + \infty$ for $m \to + \infty$; thereby, it is not of interest to me whether the sequence $(f_m)_m$ actually converges to a limit function (and in which sense) or not, but if things are easier to state, convergence can be assumed. Furthermore, let $g: G \longrightarrow G$ be a $K$-quasiconformal mapping for some (finite) $K \in [1, +\infty)$ which is non-trivial, i.e. $g \not = \operatorname{id}_G$. Now the composition $f_m \circ g: G \longrightarrow \mathbb{C}$ is defined, giving rise to consider the sequence $(f_m \circ g)_m$ of quasiconformal mappings.
My question is: Will this new sequence of quasiconformal mappings have unbounded maximal dilatation as well, i.e. is it always true that $K(f_m \circ g) \longrightarrow + \infty$ for $m \to + \infty$? As far as I know, one only has the inequality $K(f_m \circ g) \leq K(f_m) K(g)$ at hand.
Thanks in advance for any help!
Yes, this follows from the inequality you cited, the observation that $f_m = (f_m \circ g) \circ g^{-1}$, and the fact that $K(g) = K(g^{-1})$.