Max/Min modulus principle. Let $f$ be analytic function  inside and on a bounded domain $D$.
If $\Re(f)$ is constant on the boundary, then $f$ is constant in $D$.
I realized that maximum value of $\Re(f)$ must occur on the boundary, but I am not sure why minimum value of $\Re(F)$ also occurs on the boundary of $D$.
Can anyone explicitly state what Max/Min modules principle is ?
 A: Any harmonic function must attain both its maximum and minimum on the boundary of any compact set $D$.  The real part of any holomorphic function is harmonic.
If $\Re(f)$ is constant on the boundary of $D$, then $\Re(f)$ is constant on $D$ by the above.  By the Cauchy-Riemann equations, $\Im(f)$ is also constant, so $f$ is constant.

Further explanation:
$\Re(f)$ is harmonic; this follows from Cauchy-Riemann.
A basic property of harmonic functions is that, if $g$ is harmonic, and $g$ is defined in an $\epsilon$-ball around $x$, then $g(x)$ equals the average value of $g$ on $B(x,\epsilon)$ (also, the average value of $g$ on the surface of $B(x,\epsilon)$).  It follows that if $g$ attains a maximum (or minimum) at a point $x$ in the interior of $D$, then $g$ is constant in any ball around $x$.
A: Consider the analytic function $\exp(f)$. By the min/max modulus principle, $|\exp(f)| = \exp(\Re(f))$ must be min/max on the boundary of $D$. Since the real exponential is an increasing function, you get the conclusion.
