## Access

You are not currently logged in.

Access JSTOR through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Journal Article

# 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

#### Select the topics that are inaccurate.

Cancel
Preview not available

## 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.

• 285
• 286
• 287
• 288
• 289
• 290
• 291
• 292
• 293