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

Phragmén-Lindelöf theorem in a cohomological form