PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—August 1999

Notation: B = {z ∈ C: |z| < 1}, the open unit disc

1. Suppose that f is a nonconstant entire function and that |f(z)| ≤ 1 + |z| for all z. Show that f is one-to-one.

2. Let ∑_{n=0}^{∞} a_{n}x^{n} and ∑_{n=0}^{∞} b_{n}x^{n} be power series which converge for −1 < x < 1. Suppose that, when x = \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, …, then

∑_{n=0}^{∞} a_{n}x^{n} = ∑_{n=0}^{∞} b_{n}x^{n}.

Show that a_{n} = b_{n} for all n.

3. Let f be a nonconstant entire function, and define g by g(z) = f(1/z) if z ≠ 0. Prove that f is a polynomial if and only if g has a pole at 0.

4. Suppose f is analytic on a neighborhood of \overline{B} and |f(z)| < 1 when |z| = 1. Use Rouché’s Theorem to determine how many fixed points f must have in B. That is, determine how many solutions the equation f(z) = z has there.

5. By definition, an automorphism of a connected open set Ω is a one-to-one analytic function mapping Ω onto Ω.

(a) Let ϕ and ψ be automorphisms of B. Suppose there are two distinct points a and b in B such that ϕ(a) = ψ(a) and ϕ(b) = ψ(b). Show that ϕ = ψ on B.

(b) Let D be a simply connected domain in C which is not equal to C, and let ϕ and ψ be automorphisms of D. Suppose there are two distinct points a and b in D such that ϕ(a) = ψ(a) and ϕ(b) = ψ(b). Show that ϕ = ψ on D.

6. Let F denote the set of all f analytic on B such that

∫∫_{B} |f(x + iy)|^{2} dxdy ≤ 1.

Show that F is a normal family (that is, F is uniformly bounded on compact subsets of B). Hint: Fix p ∈ B. Show first that

|f(p)|^{2} ≤ \frac{1}{2π} ∫_{0}^{2π} |f(p + re^{it})|^{2} dt

for 0 < r < 1 − |p|. It follows from this (how?) that

|f(p)|^{2} ≤ \frac{1}{π(1 − |p|)^{2}} ∫∫_{|x+iy−p|<1−|p|} |f(x + iy)|^{2} dxdy,

and the result follows.