Asymptotics of the coefficient quasiconformality, and the boundary behavior of a mapping of a ball

It is shown that if a quasiconformal automorphism $f\colon B^n\to B^n$ of the unit ball in $\mathbf R^n$ $(n\geqslant 2)$ has coefficient of quasiconformality $K_f(r)=\sup\limits_{|x|\le r}k(f,x)$ in the ball of radius $r<1$ with asymptotic growth such that $\int\limits^1K(r)\,dr<\infty$, then it has a radial limit at almost every point of the boundary. This asymptotic growth of $K(r)$ is sharp in a certain sense.