Hi, I know this one is not too hard, but I've been stuck for a while: Say f is holomorphic and non-constant on the closed unit disc D(0,1), and |f|=1 on the boundary of the disk (so that, e.g., by the MVT, f maps the disk into itself) . Is it the case that f maps the disk _onto_ itself? I have thought of trying to show that the integral: ∫_D (f'(z)dz/(f(z)-a ) is non-zero , for a in the interior of D. i.e., the winding number of f(z) about any point on the disk is non-zero. But I can't see how to show this. Any Ideas? I am trying to use the fact that if f is analytic, then f is a finite product of Blaschke factors , but it still does not add up. Any ideas? Thanks in Advance.