Abstract: In this paper, we prove that the control function of the dilatation function of Beurling-Ahlfors extension is convex. Using the quasi-symmetric function ρ, we get a relatively sharp estimate of the dilatation function: D(x,y)≤17/32 (ρ(x, y) + 1) (ρ(x + y/2, y/2) +ρ(x - y/2, y/2) +2) , which improves the results before. We also show that the above result is asymptotically precise.

Key words: Beurling-Ahlfors ,  , extension;quasi-symmetric ,  , function;dilatation ,  , function;con-
trol , function