Let $C$ be $K$- quasi-circle, that is, the image $f(S^1)$, where $f : \mathbb{C} \rightarrow \mathbb{C}$ is $K$-quasiconformal.
The following criterion (by Ahlfors) for when a Jordan curve $C \subset \mathbb{C}$ is a quasi-circle is well known:
A Jordan curve $C \subset \mathbb{C}$ has bounded turning with constant $h$ if there exists a fixed constant $h > 0$ such that for any two point $z_1,z_2 \in C$, $\frac{\text{diam}(C_{z_1,z_2})}{|[z_1,z_2]|} \leq h$, here $|[z_1,z_2]| = |z_2 - z_1|$ is the euclidean length of the segment $[z_1,z_2]$ and $C_{z_1,z_2}$ is the arc of $C$ joining $z_1$ and $z_2$ with the smaller euclidean diameter.
A Jordan curve $C \subset \mathbb{C}$ is a $K(h)$-quasicircle if it is of bounded turning with constant $h$. Here $K(h)$ only depends on $h$.
I would like to know if there is a known best possible estimate for $K(h)$.