Let $D$ denote the complex unit disk and $X \subset \mathbb{C}$ some subset. Let us consider a holomorphic motion $i \colon D \times X \rightarrow \mathbb{C}$ (denoted $i\_\lambda(z)$) meaning for each fixed $\lambda \in D$ the map $i\_\lambda(\cdot)$ is injective, for each fixed $z \in X$ the map $i\_{(\cdot)}(z)$ is holomorphic and lastly $i\_0 = id\_X$. It is known that for such a motion each map $i\_\lambda$ is quasi-conformal, i.e. has bounded dilatation: $$\exists K \colon~ \sup\_{x \in X} \limsup\_{t \rightarrow \infty} \frac{\sup\_{y \in \partial B\_t(x)} ~d(f(x),f(y))}{\inf\_{z \in \partial B\_t(x)} ~d(f(x),f(z))} \leq K$$ The proof proceeds as follows:


Fix four distinct points $z\_1,\dots,z\_4 \in X$ and define $$f(\lambda) = \frac{i\_{\lambda}(z\_1)-i\_{\lambda}(z\_3)}{i\_{\lambda}(z\_1)-i\_{\lambda}(z\_2)} \cdot \frac{i\_{\lambda}(z\_2)-i\_{\lambda}(z\_4)}{i\_{\lambda}(z\_3)-i\_{\lambda}(z\_4)}$$ so that $|f(\lambda)|$ is the cross ratio of the points $(i\_\lambda(z\_1),\dots,i\_\lambda(z\_4))$. By the properties of $i$, $f$ is holomorphic and omits the values $0,1,\infty$. Thus, the image of $f$ is in $\mathbb{C} \setminus \{0,1\}$, which admits a hyperbolic metric $\rho$. As $f$ is holomorphic, $\rho(f(\lambda),f(0))\leq \rho\_D(\lambda,0)$, where $\rho\_D$ is the hyperbolic metric on $D$. In particular, the right hand side is a bound independent of the quadruple fixed in the beginning. From this it follows that $i\_\lambda$ is quasi-möbius (meaning that the cross ratios of image quarduples are bounded), which implies quasi-conformality (proved by Väisälä).


Quasi-conformality implies Hölder-continuity (sometimes called Mori's theorem, see Lectures on quasi-conformal mappings by Ahlfors). Thus, for any motion, the maps $i\_\lambda$ are Hölder continuous.


Now Shishikura claims in his paper "The Hausdorff Dimension of the Mandelbrot set and Julia sets" (<https://www.jstor.org/stable/121009?seq=1#page_scan_tab_contents>) that the inverse maps $i\_\lambda^{-1}$ are also Hölder-continuous (see page 234), but does not provide a proof.
Unfortunately, this fact is crucial for the proof of his lemma 3.2 (page 235).


Can someone explain to me how it follows? Cheers!



Hölder continuity of holomorphic motions
The answer is given in @Misha's comment. This is an extended comment. You do not need the theory of quasiconformal (quasisymmetric) maps on arbitrary sets $X$ here, because there is a stronger form of $\lambda$-lemma: every holomorphic motion of any set extends to a holomorphic motion of the whole
Riemann sphere, with $K$ estimated in terms of $|\lambda|$. Thus it is quasiconformal on the whole sphere. The inverse of a quasiconformal map is quasiconformal, without any additional restrictions, and all quasiconformal maps are Holder (also without any additional restriction; the Holder exponent is estimated by a function of $K$ alone).


