1. The problem statement, all variables and given/known data Let f=u+iv, where u(z)>v(z) for all z in the complex plane. Show that f is constant on C. 2. Relevant equations none 3. The attempt at a solution Here's my attempt (just a sketch): Since f is entire, then its components u(z), v(z) are also entire <- is this necessarily true? Since f is entire, the CR equations hold. v(z) is bounded for all z, so since v is entire and bounded, by liouville's theorem v is constant. <- kinda shaky on this one, too. Since CR equations hold, we know v_y = u_x = 0 and -v_x = u_y = 0. So, f' = u_x - iu_y = 0, so f is constant. Does this seem okay.. or am I way off base?