1. The problem statement, all variables and given/known data Let f:C->C be an entire function such that Imf(z) <= 0 for all z in C. Prove that f is constant. 2. Relevant equations Cauchy-Riemann equations?? 3. 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.