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

Abstract

In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:$ if $f(\Delta )\cap \{0,1,\infty \}=\varnothing ,$ then% A(f,Δ)\leq hL(f,\partial Δ),% 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 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_{τ\in \lbrack 0,1]}[ \frac{\sqrt{1+τ^{2}}(π+\arcsin τ)}{\mathrm{{arccot}\frac{\sqrt{1-τ^{2}}}{\sqrt{% 1+τ^{2}}}}}-τ] =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 and \lim_{n\to \infty}A(f_{n},Δ)/L(f_{n},\partial Δ)=h_{0}.

83 pages