by using Maximum AND minimum modulus principle

Let f be a nonconstant analytic function on the closed bounded region $\{ z \in C ; \mid z \mid \leq 1 \}$ .Suppose that $\mid f(z) \mid$ is constant on boundary [i.e there is $k \in R with \mid f(z) \mid =k \forall z s.t \mid z \mid =1$ prove that there is $z_0 \in C , \mid z_0 \mid \prec 1 s.t \mid f(z_0 )\mid = 0$