Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \geq 1$ such that for each $r > 0$ and $x \in U$ there exists an $s$ such that the image of an $r$-ball centered at $x$ in $U$ contains a $K^{-1}s$ ball and is contained inside of a $Ks$ ball centered at $f(x)$. Is there anything at all that I can say about $f$? I've been struggling to find any literature on this topic as the quasiregular mapping theory that one might resort to when injectivity fails in the quasiconformality setting requires differentiability.

In the context I am studying it in (a rigidity problem in dynamical systems) I do have some other knowledge about this function $f$; it is Holder and it has certain homogeneity/scaling properties that force the quasiconformality type relation above.