1. More Complex Help

I have been given this problem...

f(z) = u(x, y) + iv(x, y) is entire and |u(x, y)| < m for all points
(x, y) in the xy plane. Prove that u(x, y) is constant.

I know that since f(z) is entire if f(z) were bounded it would satisfy Liouville's theorem and it would be constant.

I have thought about this problem a lot and I am thinking I need to show this using the fact that iv(x,y) is the harmonic conjugate of u but HOW???

Any help would be greatly appreciated.

2. Originally Posted by ginafara
I have been given this problem...

f(z) = u(x, y) + iv(x, y) is entire and |u(x, y)| < m for all points
(x, y) in the xy plane. Prove that u(x, y) is constant.

I know that since f(z) is entire if f(z) were bounded it would satisfy Liouville's theorem and it would be constant.

I have thought about this problem a lot and I am thinking I need to show this using the fact that iv(x,y) is the harmonic conjugate of u but HOW???

Any help would be greatly appreciated.
HINT: What can you say about the function $g(z) = e^{f(z)}$ ?