In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:% \bar{ \Delta}\to S,$ if $f(\Delta)\cap \{ 0,1,\infty \}=\emptyset ,$ then% A(f,\Delta)\leq hL(f,\partial \Delta),% where $\Delta $ is the disk $|z|<1$ in $\mathbb{ C},$ $S$ is the unit Riemann sphere, $A(f,\Delta)$ is the area of the image of $\Delta $ and $% L(f,\partial \Delta)$ is the length of the image of $\partial \Delta $, both counting multiplicities. In this paper, we will show that the best lower bound for $h$ is the number h_{ 0}=\max_{ \tau \in \lbrack 0,1]}[ \frac{ \sqrt{ 1+\tau ^{ 2}}(\pi +\arcsin \tau)}{ \mathrm{ { arccot}\frac{ \sqrt{ 1-\tau ^{ 2}}}{ \sqrt{ % 1+\tau ^{ 2}}}}}-\tau ] =4. \allowbreak 034 159 790 \allowbreak 51..., % and this is the exact estimation, i.e. there exists a sequence of holomorphic mappings $f_{ n}:\bar{ \Delta}\to S$ such that $% f_{ n}(\Delta)\cap \{ 0,1,\infty \}=\emptyset $ and \lim_{ n\to \infty}A(f_{ n},\Delta)/L(f_{ n},\partial \Delta)=h_{ 0}.
Categories: math.CV math.DG