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# A Phragmén-Lindelöf Theorem Conjectured by D. J. Newman

W. H. J. Fuchs
Transactions of the American Mathematical Society
Vol. 267, No. 1 (Sep., 1981), pp. 285-293
DOI: 10.2307/1998584
Stable URL: http://www.jstor.org/stable/1998584
Page Count: 9
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## 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 $\lim \sup_{z\rightarrow\zeta,z \in D} |f(z)| \leqslant 1$ for $\zeta \in \partial D, \zeta \neq \infty$, then one of the following holds (a) $|f(z)| < 1 (z \in D)$, (b) $f(z)$ has a pole at $\infty$, (c) $\log M(r)/\log r \rightarrow \infty$ as $r \rightarrow \infty$. If $M(r)/r \rightarrow 0(r \rightarrow \infty)$, then (a) must hold.

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