8. Let S ? C be an open in?nite strip and f : S ? C be continuous, analytic on S , and such that…

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8.    Let S ⊆ C be an open infinite strip and f : S → C be continuous, analytic on S, and such that Re f (z) = 0 for all z ∈ ∂S. Prove that f is unbounded or constant.

(Consider the technique used in the proof of Theorem 6.3.6.) Give an example where f

is unbounded.