Complex analysis preliminary examination August 31, 2000

1. Let f be a harmonic function in the unit disk. Given that e^{f} is harmonic prove that eithr f is holomorphic, or f̄ is holomorphic.

2. Let f_{n}, n = 0, 1, ... be a sequence of functions analytic in a closed bounded domain Ω̄ with smooth boundary. Suppose that f_{n} → f_{0} as n → ∞ uniformly on compact subsets of Ω. Does this imply that

∫_{∂Ω} |f_{n}(z)|ds(z) → ∫_{∂Ω} |f_{0}(z)|ds(z),

where ds is the linear Lebesgue measure on ∂Ω. Prove, or give a counterexample.

3. If f is analytic in the unit disk Δ, continuous in Δ̄ and maps Δ into itself, prove that for every point a ∈ Δ

f^{(3)}(a) ≤ \frac{6(1 + |a|^{2})}{(1 − |a|^{2})^{3}}.

Hint: Use Cauchy formula.

4. Prove that

∫_{0}^{∞} \frac{(log x)^{2}}{1 + x^{2}} dx = \frac{π^{3}}{8} .

5. Let B be a finite Blaschke product. Prove that

\frac{1}{2πi} ∫_{|z|=1} \frac{dz}{B(z)} = B̄′(0).

6. Let f be analytic and absolutely integrable with respect to the Lebesgue area measure dA = dxdy in the unit disk Δ. Prove that for a ∈ Δ

∫_{Δ} f(z)\frac{z̄}{(1 − az̄)^{3}} dA(z) = \frac{1}{2}f′(a).

7. Let f be analytic in the closed unt disk and |f(z)| is constant for |z| = 1. Prove that arg(f(e^{iθ})) is a monotone function of θ.

8. Let f be a smooth bounded function in the unit disk Δ and F (z) be given by

F (z) = \frac{1}{π} ∫_{Δ} \frac{f(w)dA(w)}{w − z}.

Prove that F(z) is analytic outside of Δ̄ and for every z ∈ Δ

\frac{∂F}{∂z̄} = f(z).

Hint: Use Cauchy-Green formula.