Complex analysis preliminary examination
August 31, 2000
1. Let f be a harmonic function in the unit disk. Given that ef is harmonic
prove that eithr f is holomorphic, or f̄ is holomorphic.
2. Let fn, n = 0, 1, ... be a sequence of functions analytic in a closed
bounded domain Ω̄ with smooth boundary. Suppose that fn → f0 as n →
∞ uniformly on compact subsets of Ω. Does this imply that
∫
∂Ω
|fn(z)|ds(z) →
∫
∂Ω
|f0(z)|ds(z),
where ds is the linear Lebesgue measure on ∂Ω. Prove, or give a counterex-
ample.
3. If f is analytic in the unit disk ∆, continuous in ∆̄ and maps ∆ into
itself, prove that for every point a ∈ ∆
f (3)(a) ≤ 6(1 + |a|
2)
(1− |a|2)3
.
Hint: Use Cauchy formula.
4. Prove that
∫ ∞
0
(log x)2
1 + x2
dx =
π3
8
.
5. Let B be a finite Blaschke product. Prove that
1
2πi
∫
|z|=1
dz
B(z)
= ¯
B′(0).
1
2
6. Let f be analytic and absolutely integrable with respect to the Lebesgue
area measure dA = dxdy in the unit disk ∆. Prove that for a ∈ ∆
∫
∆
f(z)
z̄
(1− az̄)3
dA(z) =
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(eiθ)) is a monotone function of θ.
8. Let f be a smooth bounded function in the unit disk ∆ and F (z) be
given by
F (z) =
1
π
∫
∆
f(w)dA(w)
w − z
.
Prove that F(z) is analytic outside of ∆̄ and for every z ∈ ∆
∂F
∂z̄
= f(z).
Hint: Use Cauchy-Green formula.