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

TL;DR

This paper determines the exact optimal lower bound for the area--length ratio in Ahlfors' covering surface theory, establishing that the best constant is approximately 4.034, through explicit construction and analysis.

Contribution

It precisely calculates the maximal lower bound for the area--length ratio in Ahlfors' theory, improving understanding of extremal holomorphic mappings.

Findings

The best lower bound for the area--length ratio is approximately 4.034.

This bound is proven to be optimal via a sequence of mappings approaching it.

The exact value of the bound is given by a specific maximization formula.

Abstract

In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:% \bar{\Delta}\to S,$ 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}.