(a) Verify Theorem 3 for

(i) F(z) = z² and the rectangle 1 ≤ x ≤ 5, 2 ≤ y ≤ 4

(ii) F(z) = sin z and the unit disk

(iii) F(z) = e^z and any bounded domain.

(b) F(z) = 1 + |z| is not zero in the disk |z| ≤ 2 and has a minimum at an interior point. Does this contradict Theorem 3?

(c) F(x) = sin x (x real) has a maximum 1 at π/2. Why can this not be a maximum of |F(z)| = |sin z | in a domain containing z = π/2?

(d) If F(z) is analytic and not constant in the closed unit disk D: |z| ≤ 1 and |F(z)| = c = const on the unit circle, show that F(z) must have a zero in D.