Let f(z) be a K-quasiconformal mapping of a domain \(G_{n,r}=\{z;\quad| z| <1\}\backslash\{z=\rho \exp 2k\pi n^{- 1}i;\quad 0\leq\rho \leq r,\quad k=1,2,...,n\}\) which maps \(G_{n,r}\) into \(\{\) w; \(0<| w| <1\}\) and is normalized by \(| f(e^{i\theta})| =1\) for \(0\leq\theta \leq 2\pi.\) The bounded component of the complements of \(f(G_{n,r})\) is denoted by \(B_ f\) and the farthest point from the origin of \(B_ f\cap\{w;\quad\arg w=2k\pi n^{-1}\}, k=1,2,...,n\), denoted by \(w_ k\). The author obtains the following exact estimates: (1) Suppose \(\Phi_ n(r)=1/\Phi^{1/n}(1/r^ n),\) where \(\Phi\) (p), \(p>1\) is the Grötzsch’s regional function, then \(\Phi_ n(_{1\leq k\leq n}| w_ k|)\leq\Phi_ n^{1/k}(r). (2)\quad_{1\leq k\leq n}| w_ k| r^{-1/k}\leq 4^{(1/n)(1-1/k)}.\) Recently, the author obtained a further result: Let f(z) be a K-quasiconformal mapping of \(\{| z| <1\}\) into \(\{| w| <1\}\), normalized by \(f(0)=0,\) then \(\prod^{n}_{k=1}| f(r \exp 2k\pi n^{-1}i| r^{-1/k}\leq 4^{1-1/k}.\) Considering that \(_{1\leq k\leq n}| w_ k|\leq (\prod^{n}_{k=1}| w_ k|)^{1/n}\) and the definition of \(f_ n(r \exp 2k\pi n^{-1}i)\) is more evident than that of \(w_ k\), this estimate is an improvement of the formers.