Lemma 7.12.

i.

If \(\, f(z) = 0,\, f(z)\,\) is analytic for \(\,\left |z\right | < 1\,\) and \(\, \left |f(z)\right |< 1,\,\) then \(\, \left |f(z)\right |\leq \left |z\right |\,\) and \(\, \left |f'(0)\right |\leq 1\)

Moreover

ii.

If \(\, \left |f(z)\right | = \left |z\right |\,\) at an interior of \(\, \left |z\right | = 1\), then \(\, f(z) = ze^{i\alpha }\,\) with \(\alpha \) real.