# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Phragmén-Lindelöf theorem in a cohomological formHTML articles powered by AMS MathViewer

by Ching Her Lin
Proc. Amer. Math. Soc. 89 (1983), 589-597 Request permission

## Abstract:

The main result of this paper is as follows. Given functons ${\phi _1}(\varepsilon ), \ldots ,{\phi _\nu }(\varepsilon )$ which are holomorphic in sectors ${S_1}, \ldots ,{S_p}$, respectively, where ${S_1} \cup \cdots \cup {S_\nu } = \{ \varepsilon :|\arg \varepsilon | < \pi /2\alpha ,0|\varepsilon | < \rho \}$ for $\alpha > 1$, $\rho > 0$, set ${\phi _{jk}} = {\phi _j} - {\phi _k}$ if ${S_j} \cap {S_k} \ne \emptyset$. Then $\{ {\phi _{jk}}\}$ satisfy cocycle conditions ${\phi _{jk}} + {\phi _{kl}} = {\phi _{jl}}$ whenever ${S_j} \cap {S_k} \cap {S_l} \ne \emptyset$. In addition to the conditions $|{\phi _1}| < {M_0}$ and $|{\phi _\nu }| < {M_0}$ on the two rays of the boundary $({\text {i}}{\text {.e}}{\text {. }}\arg \varepsilon = \pi /2\alpha )$, and $|{\phi _j}(\varepsilon )| \leqslant A\exp (c/|\varepsilon |)$ in ${S_j}$ for some positive numbers $A$ and $c$, $j = 1,2, \ldots ,\nu$, if the $\{ {\phi _j}\}$ satisfy the conditions $\{ {\phi _{jk}}\} < {M_0}$ on ${S_j} \cap {S_k}( \ne \emptyset )$, then we get $|{\phi _j}| < M$ on $S$, $j = 1,2, \ldots ,\nu$. (From the cohomological point of view, we can get global results for ${\phi _j}$,oce the local data on cocycles is known.)
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