TITLE: What might I use to show that an entire function with positive real parts is constant?
QUESTION [9 upvotes]: So the question asks me to prove that an entire function with positive real parts is constant, and I was thinking that this might somehow be related to showing an entire bounded function is constant (Liouville's theorem), but are there any other theorems that might help me prove this fact?

REPLY [6 votes]: Well, can't we just say that, since $-f(z)$ is entire, $e^{-f(z)}$ is also entire, and if we write
$f(z) = u(z) + iv(z), \tag{1}$
where $u(z)$, $v(z)$ are the (harmonic) real and imaginary parts of $f(z)$ (so that $u(z) = Re \; f(z)$), then
$\vert e^{-f(z)} \vert = \vert e^{-u(z) - iv(z)} \vert = \vert e^{-u(z)} \vert \vert e^{-iv(z)} \vert = e^{-u(z)}, \tag{2}$
since
$e^{-u(z)} > 0 \tag{3}$
and
$\vert e^{-iv(z)} \vert = \vert \cos (-v(z)) + i\sin (-v(z)) \vert = 1; \tag{4}$
but $-u(z) < 0$ by hypothesis; thus $e^{-u(z)} < 1$, whence $e^{-f(z)}$ is a bounded entire function, hence constant; hence $f(z)$ must itself be a constant. QED.
Can't we just say that?  I think we can!