Question

Prove that an analytic function with constant real part is constant.

Answer 1

I want to prove that if f is an analytic function for which If is constant, this implies that f itself is constant.

So to start off, it's not given that the function is entire or anything, ruling out Liouville. Rather, I'm guessing the maximum modulus theorem will prove useful here (?).

If I were to show using that the real part of f was constant, proving the statement would be easy (just take the absolute value and investigate ef), but in this case I'm none the wiser from that. Should I use the same parametrization ef and apply the Cauchy-Riemann equations somewhere? Any suggestions welcome.