1
$\begingroup$

Possible Duplicate:
Liouville's theorem problem

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be entire and suppose $\exists M \in\mathbb{R}: $Re$(f(z))\geq M$ $\forall z\in\mathbb{C}$. How would you prove the function is constant?

I am approaching it by attempting to show it is bounded then by applying Liouville's Theorem. But have not made any notable results yet, any help would be greatly appreciated!

  • 0
    @ZevChonoles: Thanks, I think that was the right decision, it's good for my reference2011-10-30

2 Answers 2

6

Consider the function $\displaystyle g(z)=e^{-f(z)}$. Note then that $\displaystyle |g(z)|=e^{-\text{Re}(f(z))}\leqslant \frac{1}{e^M}$. Since $g(z)$ is entire we may conclude that it is constant (by Liouville's theorem). Thus, $f$ must be constant.

  • 1
    I think "Consider the function $\displaystyle g(z)=e^{-ff(z)}$" should be replaced by "Consider the function $\displaystyle g(z)=e^{-f(z)}$."2013-05-27
0

Since this seems like homework (if it is, you should use the homework tag), I will only give a hint.

Think about what the image domain of the function will look like. Can you postcompose $f$ with a simple holomorphic function on this domain (e.g. a Möbius transformation) such that the new function $g$ you obtain is bounded?