Phragmén-Lindelöf theorem in a cohomological form

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.)