A Phragmén-Lindelöf theorem conjectured by D. J. Newman

Abstract:

Let $D$ be a region of the complex plane, $\infty \in \partial D$. If $f(z)$ is holomorphic in $D$, write $M(r) = {\sup _{|z| = r, z \in D}}|f(z)|$. Theorem 1. If $f(z)$ is holomorphic in $D$ and $\lim {\sup _{z \to \zeta , z \in D}}|f(z)| \leqslant 1$ for $\zeta \in \partial D$, $\zeta \ne \infty$, then one of the following holds (a) $|f(z)| < 1(z \in D)$, (b)$f(z)$ has a pole at $\infty$, (c) $\log M(r)/\log r \to \infty$ as $r \to \infty$. If $M(r)/r \to 0(r \to \infty )$, then (a) must hold.