It is known for some time that there exists an energy-minimal diffeomorphism between two doubly-connected domains $\Omega$ and $D$ provided that $\mathrm{ Mod}(\Omega)\le \mathrm{ Mod}{ D}$ and that diffeomorphism is harmonic \cite{ tedi}. In this note we give a short proof of the fact that for given annuli $\Omega$ and $D$ satisfying the condition $\mathrm{ Mod}(\Omega)\le \mathrm{ Mod}(D)$ there exist a $K-$quasiconformal harmonic diffeomorphism $f:\Omega\onto D$, where $K=K(\tau)$, $\tau=\mathrm{ Mod}(\Omega)/ \mathrm{ Mod}(D)$ and $\lim_{ \tau\to 1}K(\tau)=1$.
Categories: math.CV