The best bound of the area--length ratio in Ahlfors Covering surface theory (I)

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 $\mathrm{\Delta }$ is the disk $\mathrm{\mid }z\mathrm{\mid }<1$ in $\mathbb{C},$ $S$ is the unit Riemann sphere, $A(f,\mathrm{\Delta })$ is the area of the image of $\mathrm{\Delta }$ and $$ is the length of the image of $\mathrm{\partial }\mathrm{\Delta }$, 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 $f_n:\bar{\mathrm{\Delta }}\to S$ 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}.