$f$ is holomorphic on an open unit disk $D$. If $arg(f)$ is constant in $D$, then $f$ is constant in $D$
I don't know where to start:
I know the Cauchy Riemann Equations (these must be satisfied for complex differentiability) : $u_x=v_y$ and $v_x=-u_y$
I know the open mapping theorem - every open set is mapped to an open set
I know the maximum modulus theorem - if $f$ is non-constant, then it can't attain a maximum in $D$.