Complex Analysis and Linear Algebra

Exercise 7. The family of mappings introduced here plays an important role in complex analysis. These mappings, sometimes called Blaschke factors, will reappear in various applications in later chapters.

(1) Let z, w be two complex numbers such that zw 6= 1. Prove that     w − z 1 − wz     < 1 if |z| < 1 and |w| < 1, and also that     w − z 1 − wz     = 1 if |z| = 1 or |w| = 1.

(2) Prove that for a fixed w in the unit disc D, the mapping F : z 7→ w − z 1 − wz satisfies the following conditions

(a) F maps the unit disc to itself (that is, F : D → D), and is holomorphic.

(b) F interchanges 0 and w, namely F(0) = w and F(w) = 0.

(c) |F(z)| = 1 if |z| = 1.

(d) F : D → D is bijective.