Question: 4. Let f be an entire function such that |f (z)| eRe(z) for all z C. Prove that f (z) = cez for some constant

4. Let f be an entire function such that |f (z)| eRe(z) for all z C. Prove that f (z) = cez for some constant c C. Proof. |f (z)| |eRe(z)| = |ez | so | f (z) ez | 1 and f (z) ez is entire so by Liouville's theorem f (z) ez = c for some constant c C

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