 Solution

Write the holomorphic function as $f=u+iv$, with $u^2+v^2=C$. If $C=0$ then $f=0$. If $C>0$, differentiating the constant modulus with respect to both coordinates, and using the Cauchy-Riemann equations $u_y=-v_x$, $v_y=u_x$, givesThe coefficient matrix has determinant $-(u^2+v^2)=-C\ne 0$, so $u_x=v_x=0$. The Cauchy-Riemann equations then also give $u_y=v_y=0$. Along a straight segment inside the disk, both real components have zero derivative and hence are constant. Thus $f$ is constant throughout the disk.