How can i proof an analytic function cannot have a constant absolte value without reducing to a constant?

To prove that an analytic function cannot have a constant absolute value without being a constant function, consider that if ( f(z) ) is analytic and ( |f(z)| = c ) (a constant) in a region, then ( f(z) ) must have a constant argument, implying that ( f(z) ) is of the form ( c e^{i\theta} ). By the Cauchy-Riemann equations, the derivative ( f'(z) ) must be zero in that region, which means ( f(z) ) is constant throughout the region. Thus, any non-constant analytic function cannot maintain a constant absolute value.