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I am studing for exams and am stuck on this problem.

Suppose $f$ is an entire function s.t. $f(z) =f(z+1)$ and $|f(z)| < e^{|z|}$. Show $f$ is constant.

I've deduced so far that: a) $f$ is bounded on every horizontal strip b) for every bounded horizontal strip of length greater than 1 a maximum modulus must occur on a horizontal boundary.

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    So we need to show that $f$ is bounded and entire. We are given that it is entire. It seems that $|f(z)| < e^{|z|}$ and $|f(z+1)|< e^{|z|}$. Then $|f(z+2)| < e^{|z+1|}$.2010-12-15
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    Yes. So as you said $f$ is bounded on horizontal strips. I don't know how to show boundedness for the vertical strips2010-12-15
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    I think a bit of heavier machinery may be in order. You may want to estimate $\Delta\log|f|$ in the ball or radius $R$ and let $R\to\infty$. (Some familiarity with Jensen's formula may help.)2010-12-15

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