Entire Function with Negative Imaginary Values: Proving Constantness

[variety]

Homework Statement

Let f:C->C be an entire function such that Imf(z) <= 0 for all z in C. Prove that f is constant.

Homework Equations

Cauchy-Riemann equations??

The Attempt at a Solution

I don't know why I haven't been able to get anywhere with this problem. I feel like I have to use the fact that Imf(z) is harmonic or satisfies the Cauchy-Riemann equations, or something like that. And then somehow show that f is bounded. From there I just apply Liouville's Theorem. But I just need a slight push in the right direction. I mean, if Imf(z) <= 0 for all z, what does that say about its derivatives? This is really frustrating.

 

[yyat]

Hint: Consider the function 1/(f(z)-i). Is it bounded?