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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Phragmén-Lindelöf theorem in a cohomological form


Author: Ching Her Lin
Journal: Proc. Amer. Math. Soc. 89 (1983), 589-597
MSC: Primary 30A10
MathSciNet review: 718979
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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 :\vert\arg \varepsilon \vert < \pi /2\alpha ,0\vert\varepsilon \vert < \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 $ \vert{\phi _1}\vert < {M_0}$ and $ \vert{\phi _\nu }\vert < {M_0}$ on the two rays of the boundary $ ({\text{i}}{\text{.e}}{\text{. }}\arg \varepsilon = \pi /2\alpha )$, and $ \vert{\phi _j}(\varepsilon )\vert \leqslant A\exp (c/\vert\varepsilon \vert)$ 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 $ \vert{\phi _j}\vert < 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0718979-7
PII: S 0002-9939(1983)0718979-7
Keywords: Asymptotic theory, inequalities in the complex domain
Article copyright: © Copyright 1983 American Mathematical Society