= 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> $u_y=-v_x$, $v_y=u_x$, gives
$$
u u_x+v v_x=0,\qquad v u_x-u v_x=0.
$$
The coefficient matrix has <determinant> $-(u^2+v^2)=-C\ne0$, 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. \b[Thus $f$ is constant throughout the disk.]
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