Basic Complex Analysis: Cauchy Riemann

[snipez90]

Homework Statement

Let f be a holomorphic function in the unit disc D1 whose real part is constant.
Prove that the imaginary part is also constant.

Homework Equations

Cauchy Riemann equations

The Attempt at a Solution

Hi guys, I'm working through the basics again. I think here we just need to use Cauchy Riemann and the fact that the unit disc D1 is connected?

 

[gomunkul51]

C-R equation should suffice given the real part of an analytic function is constant.

 

[snipez90]

Yes I thought so, thanks. I knew this was a basic fact even though there are similar sounding problems that often use other tools. Anyways I guess the details are just an application of the mean value theorem.