She quantitative version of the Kado theorem

The main aim of the paper is to prove the following result.
Theorem. Let $\Gamma$ be a $k$–quasiconformal circle, $L$ a Jordan curve (not necessarily quasiconformal). Suppose that $f$ maps $\operatorname{ext} L$ onto $\operatorname{ext} \Gamma$ quasiconformally and that $f(\infty)=\infty$, $f'(\infty)>0$. Suppose further that there is a horaeomorphism $\chi\colon L\to\Gamma$ such that
$$ |\chi(\zeta)-\zeta|\leqslant\varepsilon,\quad\zeta\in\Gamma,\quad0<\varepsilon\leqslant1. $$
Then there exist numbers $\alpha=\alpha(k)>0$ and $A=A(k)$ such that
$$ |f(\chi(\zeta))-\zeta|\leqslant A\varepsilon^\alpha,\quad\zeta\in\Gamma. $$