PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—June 1999

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

1. Let f and g be analytic functions on B so that f(z)g(z) = 0 for all z ∈ B. Show that either f(z) = 0 for all z ∈ B or g(z) = 0 for all z ∈ B.

2. Show that there does not exist a sequence of (analytic) polynomials P_{n}(z) which converges to 1/z uniformly on {z: |z| = 1}.

3. Let γ be the simple closed curve defined by γ(t) = e^{2πit} for 0 ≤ t ≤ 1, and let f be entire. (a) Suppose there exists a positive integer n such that

∫_{γ} \frac{f(z)}{(z − α)^{n}} dz = 0

for all α ∈ B. Prove that f is a polynomial.

(b) Suppose that for each α ∈ B there exists a positive integer n(α) such that

∫_{γ} \frac{f(z)}{(z − α)^{n(α)}} dz = 0.

Prove that f is a polynomial.

4. Suppose that f is analytic on C and that |Imf| ≤ |Ref|. Show that f is constant.

5. Determine the number of roots of the equation

2z^{5} − 6z^{2} + z + 1 = 0

in the region {z: 1 < |z| < 2}.

6. Let F denote the family of functions f analytic on B so that f(0) = 0 and

|f′(z)| ≤ \frac{1}{1 − |z|}

for all z ∈ B. Show that F is a normal family.