Boundedness for Bloch functions (Q1085303)

The author considers a crescent G in the unit disc bounded by two circles \[ \{| z| =1\}\cup \{| z-x| =1-x\}\quad for\quad 0\leq x\leq. \] He proves that if f is in \(L^ p(G,dx dy)\) the closure of the (analytic) polynomials on G, and if \(| f(z)| \leq M\) on G then there is a function F, analytic on the unit disc, \(F(z)=f(z)\) for all \(z\in G\) and \(| F(z)|\) is bounded. He also proves the following. Let E be a finite subset of the unit circle and let f be an arbitrary Bloch function on the unit disc. Assume lim sup\(| f(z)| \leq K\) as \(z\to a\), all a in the unit circle except E, then \(| f|\) is bounded on the unit disc. The latter result was proven in a more general setting and in more generality by \textit{R. D. Berman} and \textit{W. S. Cohn}, ''Phragmén-Lindelöf theorems for subharmonic functions on the unit disc'' to appear).