Schlicht regions for entire and meromorphic functions (Q1293559)

Let \(X\) be a Riemann surface, \(f:X\to\overline{\mathbb C}\) be a meromorphic function, \(M\) be some class of meromorphic functions on \(X\). Let \(p\in X\) and \(d_f(p)\) be the radius of the largest spherical disc centered at \(f(p)\) in which the function \(f^{-1}\) is holomorphic. The Bloch constants are defined as follows: \[ B_f=\sup_{p\in X}d_f(p),\qquad B(X,M)=\inf_{f\in M}B_f . \] The evaluation of \(B(X,M)\) when \(X={\mathbb C}\), \(M\) being the class of all meromorphic functions is an open problem. The authors prove that \(B({\mathbb C},E)=\frac\pi{2}\) where \(E\) is the class of all entire functions. They investigate other cases as well.