 Constant-modulus holomorphic function

A holomorphic function with constant modulus on a connected domain is constant. If $h=u+iv$ and $u^2+v^2=c^2>0$, differentiation and the Cauchy-Riemann equations give $u{u}_x+v{v}_x=0$ and $-u{v}_x+v{u}_x=0$. Their coefficient determinant is $-c^2$, so $u_x=v_x=0$, and all first derivatives vanish. When $c=0$, the conclusion is immediate. Consequently two nonvanishing holomorphic functions with equal moduli differ by a single constant phase on a connected domain.