A quasiconformal analog of Carathéodory's criterion for the Möbius property of mappings

In 1937, Carathéodory proved that every injective mapping $f\colon G\to f(G)\subset\overline{\mathbf C}$ of a domain $G\subset\overline{\mathbf C}$, taking circles to circles, is Möbius. The present article shows that if each injective mapping takes circles onto $k$-quasicircles then it is $K$-quasiconformal with $K\le k+\sqrt{k^2-1}$.