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The precise bound for the area-length ratio in Ahlfors’ theory of covering surfaces. (English) Zbl 1281.30021
A key result in Ahlfors’ theory of covering surfaces is that for any surface \(\Sigma\) over \(\overline{\mathbb{C}} \setminus \{ 0,1,\infty \}\), there exists a constant \(h\) such that the isoperimetric inequality \[ A(\Sigma) \leq h L(\partial \Sigma) \] holds, where \(A,L\) denote area and length weighted according to multiplicity, respectively. This follows from an interpretation of the Second Fundamental Theorem of Nevanlinna Theory, and has further important consequences for complex analysis.
In the paper under review, the author obtains the smallest possible value of \(h\), given by \[ h_0 = \max_{\theta \in [0,\pi/2]} \frac{ (\pi + \theta )\sqrt{1+\sin^2\theta}}{\arctan \left ( \frac{ \sqrt{1+\sin^2\theta}}{\cos\theta} \right ) }. \] This is shown to be sharp by explicit construction of surfaces \(\Sigma_n\) with \[ \lim_{n\to \infty} \frac{A(\Sigma_n)}{L(\partial \Sigma_n)} = h_0. \] In this well-written paper, the author proceeds by establishing the inequality for a certain family of surfaces \(\mathbb{F}^*\), and then showing that this family well-approximates any other such surface in the sense of the quotient \(A/L\).

MSC:
30D20 Entire functions of one complex variable, general theory
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45 Normal functions of one complex variable, normal families
51M25 Length, area and volume in real or complex geometry
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