On a theorem of Phragmén-Lindelöf for analytic functions (Q2726134)

The following theorem of Phragmén-Lindelöf type is proved. NEWLINENEWLINENEWLINETheorem. Let \(G \subset \mathbb C\) be an unbounded domain with \(0 \in G\), and let \(f\) be an analytic function in \(G\) such that \(|f|\leq c\) on \(\partial G\) for some positive constant \(c\). Then there holds one of the following alternatives: (1) \(|f(z)|\leq c\) for all \(z \in G\). (2) There exist positive constants \(\alpha\), \(c_1\) and \(c_2\) such that \(M(R) \geq c_1\exp{(c_2R^\alpha)}\) for all \(R \geq R_0 > 0\), where \(M(R) = \sup\{|f(z)|: z \in G,\;|z|\leq R\}\). The constant \(\alpha\) only depends on \(G\) but not on \(f\). NEWLINENEWLINENEWLINEThe formulation of the result it is not quite clear concerning the hypothesis \(|f|\leq c\) on \(\partial G\). Does it mean that \(f\) has a continuous extension to \(\overline{G}\) or more generally that \(\limsup_{z\to\zeta}{|f(z)|} \leq c\) for all \(\zeta \in \partial G\)? The proof uses the Wiener capacity.