Math 534 Homework #3
Autumn 2008
Let D={z:|z|<1}.
1. Prove that if fis a two-to-two (one-to-one) analytic map of an open set Ω onto f(Ω) and if
zn∈Ω→∂Ω, then f(zn)→∂f (Ω), in the sense that f(zn) eventually lies outside each compact
subset of f(Ω). Another way to state the problem is to view the sets as lying on the Riemann
sphere, so that the boundary can include the North Pole (the point at “∞”).
2. Prove that ϕis a one-to-one analytic map of Donto Dif and only if
ϕ(z) = cz−a
1−¯az ,
for some constants cand a, with |c|= 1, and |a|<1. What is the inverse map?
3. For z, w ∈D, define
ρ(z, w) =
z−w
1−¯zw
.
a. Suppose fis analytic on Dand maps Dinto D. Show that ρ(f(z), f (w)) = ρ(z, w) if and only
if fis a one-to-one analytic map of Donto D.
b. Show that ρis a metric on D.
c. Prove the “world’s greatest equality”: for z, w ∈D,
1−ρ(z, w)2=(1 − |z|2)(1 − |w|2)
|1−¯zw|2,
d. Prove that ρ(|z|,|w|)≤ρ(z, w)≤ρ(|z|,−|w|).
4. Suppose fis bounded and analytic in the right half-plane {z: Rez > 0}, and lim supz→iy |f(z)| ≤
Mfor all iy on the imaginary axis. Prove |f(z)| ≤ Mon the right half-plane. Check that f(z) = ez
satisfies all the hypotheses above, except for boundedness, and fails to be bounded in the right
half-plane. Hint: consider f(z)/(1 + εz) for ε > 0 small.
5. A function f(z) on the complex plane is called “doubly periodic” if there are two complex
numbers w1and w2which do not lie on the same line through the origin such that f(z+w1) =
f(z+w2) = f(z) for all complex numbers z. The numbers w1and w2are called “periods” of f.
Prove that the only entire doubly periodic functions are constant.
