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Here is my problem:

Let $f(z)$ be an entire function such that $|f '(z)| < |f(z)|$ for all $z \in \mathbb{C}$, Show that there exists a constant A such that $|f(z)| < A*e^{|z|}$ for all $z \in \mathbb{C}$. I am trying to use Liouville's theorem to prove and try to set $g(z)=|f '(z)|/|f(z)| < 1$ and then g(z) is constant. I am not sure if my thinking is right and how to prove this problem? Thanks

  • 4
    You want to instead set $g(z) = f'(z) / f(z)$ (no absolute value). $g(z)$ is entire (why?). $g(z)$ is bounded (why?). Apply Liouville's theorem. (BTW, you can also prove this using real-variable methods by just integrating from the origin in the radial direction. See [Gronwall's inequality](http://en.wikipedia.org/wiki/Gronwall%27s_inequality).)2011-04-18
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    The reason why Willie suggests $f'/f$ and not your $g$ is that your function is not analytic, so Liouville' stheorem doesn't apply.2011-04-18

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