Suppose that $f(z)$ is an entire function and $ |f(z)| \le e^x \ (z = x + iy) $ throughout the complex plane. What can be said about $ f(z) $?
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1It feels like that $ f(z) $ is either exponential or constant zero. Is that true? – Aug 10 '13 at 23:55
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Since $\left|e^z\right| = e^x$, we have: $$ \left|\frac{f(z)}{e^z}\right| \le 1 $$
It follows that $f(z)/e^z$ is entire and bounded, hence constant by Liouville's theorem. Therefore, $f(z) = \lambda e^z$ for some $\lambda \in \mathbb C$, $|\lambda| \le 1$.
Ayman Hourieh
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2+1. As a nice generalization, if $f$ and $g$ are entire with $|f(z)| \leq |g(z)|$ for all $z$, then $f$ is a scalar multiple of $g$. – Aug 11 '13 at 00:04
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@T.Bongers Indeed. This answer here contains a statement and proof of this generalization. – Ayman Hourieh Aug 11 '13 at 00:11