A new explicit bound in Landau's theorem (Q1202838)

The theorem of Landau in question relates to functions \(f(z)\) regular in \(| z |<1\) which do not take the values 0 and 1 \(f(z) = a_ 0 + a_ 1z + \cdots\). It can be expressed as \[ | a_ 0 | \leq 2 | a_ 0 | \biggl\{ \bigl | \log | a_ 0 | \bigr | + K \biggr\} \] where \(K\) is a numerical constant. Based on the method of the reviewer [Can. J. Math. 7, 76-82 (1955; Zbl 0064.073)], \textit{Lai} [Science in China 21, 495 (1978)], \textit{J. A. Hempel} [J. Lond. Math. Soc., II. Ser. 20, 435-445 (1979; Zbl 0423.30005)] and the reviewer [Can. J. Math. 33, 559-562 (1981; Zbl 0478.30001)] independently (and given the vagaries of publication essentially simultaneously) showed that the precise value of \(K\) is \((1/4 \pi^ 2) \Gamma ({1 \over 4})^ 4\). The author obtains a bound for \(| a_ 1 |\) where the constant \(K\) is replaced by a function of \(a_ 0\) specifically \[ | a_ 1 | \leq 2 | a_ 0 | \biggl\{ \bigl | \log | a_ 0 | \bigr | + (1/4\pi^ 2) \Gamma ({1\over 4})^ 4 - m{\mathcal R} (a^ \varepsilon_ 0 + 1) \biggr\} \] with \(\varepsilon = \pm 1\) according as \(| a_ 0 | \leq 1\) or \(| a_ 0 | > 1\) and \(m\) is a constant \(\geq \cdot 04\). The method is to obtain bounds for the Poincaré metric for the sphere punctured at \(0,1,\infty\).