Phragmén-Lindelöf theorem in a cohomological form

Author:
Ching Her Lin

Journal:
Proc. Amer. Math. Soc. **89** (1983), 589-597

MSC:
Primary 30A10

DOI:
https://doi.org/10.1090/S0002-9939-1983-0718979-7

MathSciNet review:
718979

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Abstract | References | Similar Articles | Additional Information

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.)

- Ching Her Lin,
*The sufficiency of the Matkowsky condition in the problem of resonance*, Trans. Amer. Math. Soc.**278**(1983), no. 2, 647–670. MR**701516**, DOI https://doi.org/10.1090/S0002-9947-1983-0701516-5 - Walter Rudin,
*Real and complex analysis*, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR**0344043** - Yasutaka Sibuya,
*A theorem concerning uniform simplification at a transition point and the problem of resonance*, SIAM J. Math. Anal.**12**(1981), no. 5, 653–668. MR**625824**, DOI https://doi.org/10.1137/0512057

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Keywords:
Asymptotic theory,
inequalities in the complex domain

Article copyright:
© Copyright 1983
American Mathematical Society