1
$\begingroup$

$f$ be analytic such that $\Re(f)\ge 0$ then

  1. $\Im(f)=$ constant

  2. $\Im(f)\ge 0$

  3. $f=$ constant

  4. $\Re(f)=|z|$

what I have done is if $f=u+iv$ then consider, $g(z)=e^{u+iv}$ then $|g|=e^{u}$ as $g$ never vanishes so $g$ must be constant so 3 is correct in my guess. 2 is not necesarily true also 4, please help for rigoriousness.

  • 1
    $G(z) =\frac{1}{f(z)+1}$ is entire and bounded2012-06-15

1 Answers 1

4

Then here is a sketch solution:

Put $g = e^{-f}$, and consider $|g|= e^{-u}\le 1$, so that $g$ is bounded in all of $\mathbb{C}$. Then use Liouville's Theorem.

Added: Note also that this only proves that $g$ is constant in $\mathbb{C}$, and you will need to do a bit more work (e.g. calculate $g'$) to show that $f$ is constant in $\mathbb{C}$.

  • 0
    Yes, that is right.2012-06-15