Here's another approach to the scalar-valued case, which looks as though it might generalise more easily to the matrix-valued case. Since the unit disk is simply connected, we can define log(f) in such a way that it is continuous on and analytic in the interior. Use a conformal map taking the unit disk to the upper half-plane to transport log(f) to a bounded analytic function on . Then i*log(f) will be real on the real axis, so by the Schwarz reflection principle it extends to a bounded entire (therefore constant) function.