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Homework Assigned on April 21, 2009
due April 28, 2009
Problem 1 (from Stein & Shakarchi, p.259, #8). Let f be an injective holo-
morphic function in the unit disk, with f (0) = 0 and f
′ (0) = 1. If we write
f (z) = z + a 2 z
2
3
n
then Bieberbach conjectured that |an| ≤ n for all n ≥ 2, which was proved
by deBranges. This problem outlines an argument to prove the conjecture
under the additional assumption that the coefficients an are real.
(a) Let z = re
iθ with 0 < r < 1, and show that if v(r, θ) denotes the imaginary
part of f
re
iθ
, then
anr
n
π
∫ (^) π
0
v(r, θ) sin nθ dθ.
(b) Show that for 0 ≤ θ ≤ π and n = 1, 2 , · · · , we have | sin nθ| ≤ n sin θ.
(c) Use the fact that an ∈ R to show that f (D) is symmetric with respect to
the real axis, and use this fact to show that f maps the upper half-disk into
either the upper or lower part of f (D).
(d) Show that for r small,
v(r, θ) = r sin θ [1 + O (r)] ,
and use the previous part to conclude that v(r, θ) sin θ ≥ 0 for all 0 < r < 1
and 0 ≤ θ ≤ π.
(e) Prove that |anr
n | ≤ nr, and let r → 1 to conclude that |an| ≤ n.
(f) Check that the function
f (z) =
(1 − z)
2
satisfies all the hypotheses and that |an| = n for all n.
Problem 2 (from Stein & Shakarchi, p.259, #9). Gauss found a connection
between elliptic integrals and the familiar operations of forming arithmetic
and geometric means. We start with any pair (a, b) of numbers such that
a ≥ b > 0, and form the arithmetic and geometric means of a and b, that is,
a 1 =
a + b
and b 1 = (ab)
1 (^2).
We then repeat these operations with a and b replaced by a 1 and b 1. Iterating
this process provides two sequences {an} and {bn} where an+1 and bn+1 are
the arithmetic and geometric means of an and bn respectively.
(a) Prove that the two sequences {an} and {bn} have a common limit. This
limit, which we denote by M (a, b), is called the arithmetic-geometric mean
of a and b.
Hint: Show that
a ≥ a 1 ≥ a 2 ≥ · · · ≥ an ≥ bn ≥ · · · ≥ b 2 ≥ b 1 ≥ b
and
an − bn ≤
a − b
n
(b) Gauss’s identity states that
M (a, b)
π
∫ π 2
0
dθ
( a^2 cos^2 θ + b^2 sin
2 θ
2
To prove this relation, show that if I(a, b) denotes the integral on the right-
hand side, then it suffices to establish the invariance of I, namely,
(∗) I(a, b) = I
a + b
, (ab)
1 2
Then, observe that the connection with the elliptic integrals takes the form
I(a, b) =
a
K(k) =
a
0
dx √ (1 − x
2 )(1 − k
2 x
2 )
where k
2 = 1 −
b^2 a^2
, and that the relation (∗) is a consequence of the identity
K(k) =
1 + ˜k
K
1 − ˜k
1 + ˜k
for ˜k
2 = 1 − k
2 and 0 < ˜k < 1.
converges uniformly on compact subsets of Ω and also show that
∑^ ∞
n=
fn(z)fn(ζ)
converges on compact subsets of Ω × Ω with (z, ζ) ∈ Ω × Ω by using
q ∑
j=p
fn(z)fn(ζ)
2
q ∑
j=p
|fn(z)|
2
q ∑
j=p
|fn(ζ)|
2
The Bergman kernel of Ω is defined as
KΩ(z, ζ¯) =
∞ ∑
n=
fn(z)fn(ζ)
when {fn}n∈N is an orthonormal basis of the Hilbert space of all square-
integrable holomorphic functions on Ω. Verify that the Bergman kernel is
independent of the orthonormal basis {fn}n∈N chosen.
Hint: Use approximation by the following analog with a finite number of
terms. Let ξj , ξ
′ j , ηj^ , η
′ j (for 1^ ≤^ j^ ≤^ k) be complex numbers related by
ξj =
∑^ m
k=
cjkηk,
ξ
′ j =
∑^ m
k=
cjkη
′ k
for 1 ≤ j ≤ m, where the matrix (cjk)
m j,k=
is a unitary matrix. Then
∑^ m
j=
ξj ξ
′ j =
∑^ m
j=
∑m
k=
cjkηk
∑m
`=
cj`η
′ `
m ∑
k,`=
m ∑
j=
cjkcj`
ηj η
′ ` =
m ∑
j=
ηj η
′ j.
(b) (Transformation of Bergman Kernel Under Biholomorphic Map) Let Ω
and D be two domains in C, each either bounded or simply connected not
equal to C. Let Φ : Ω → D be a bibolomorphic map between Ω and D.
Verify that
KΩ(z, ζ¯) = KD
Φ(z), Φ(ζ)
′ (z)Φ′(ζ)
for z, ζ ∈ Ω, where Φ
′ means the first-order derivative of the holomorphic
function Φ on Ω. [Hint: if {fn(w)}n∈N is an orthonormal family of holomor-
phic functions on D and is a basis of the Hilbert space of all square-integrable
holomorphic functions on D, then {fn (Φ(z)) Φ
′ (z)}n∈N is orthonormal fam-
ily of holomorphic functions on Ω and is a basis of the Hilbert space of all
square-integrable holomorphic functions on Ω.]
(c) (Bergman Kernel of the Unit Disk) Verify that the Bergman kernel KD(z, ζ¯)
of the open unit disk D is given by
(†) KD(z, ζ¯) =
π
1 − z ζ¯
[Hint: the functions
n + 1
π
z
n for n ∈ N ∪ { 0 }
form an orthonormal basis of the Hilbert space of all square-integrable holo-
morphic functions on the open unit disk D.]
(d) (Riemann Mapping in Terms of Bergman Kernel) Let Ω be a simply
connected domain in C not equal to C. Let z 0 ∈ Ω and Φ : Ω → D be the
biholomorphic map such that Φ (z 0 ) = 0 and Φ
′ (z 0 ) = α > 0. Let
Ψ(z) =
∂ ∂ ζ¯
KΩ(z, ζ¯)
KΩ(z, ζ¯)
ζ=z 0
By taking
∂ ∂ ζ¯
of the logarithm of
() KΩ(z, ζ¯) = KD
Φ(z), Φ(ζ)
′ (z)Φ′(ζ)
at the point ζ = z 0 , show that
Φ(z) = α
Ψ (z) − Ψ (z 0 )
Ψ′^ (z 0 )
and letting n → ∞, derive the formula
π cot πz =
z
n∈Z−{ 0 }
z − n
n
(b) By differentiating the expression in Part(a), derive the formula
∑^ ∞
m=−∞
(m + τ )
2
π
2
sin
2 (πτ )
(c) Setting τ =
1 2
in Part(b), deduce that
m≥ 1 , m odd
m
2
π
2
and
m≥ 1
m
2
π
2
(d) Similarly, using
∞ ∑
m=−∞
(m + τ )^4
deduce that ∑
m≥ 1 , m odd
m
4
π
4
and
m≥ 1
m
4
π
2
Problem 7 (from Stein & Shakarchi, p.281, #2). Let τ ∈ C with Im τ > 0
and Λ = Z + Zτ and Λ
∗ = Λ − { 0 }. Define
P(z) =
z
2
ω∈Λ∗
[
(z + ω)
2
ω
2
]
Show that
P(z) = c + π
2
∞ ∑
m=−∞
sin
2 ((z + mτ )π)
where c is an appropriate constant.
Problem 8 (from Stein & Shakarchi, p.281, #3). Let τ ∈ C with Im τ > 0
and Λ = Z + Zτ and Λ
∗ = Λ − { 0 }. Define
P(z) =
z
2
ω∈Λ∗
[
(z + ω)
2
ω
2
]
Define
g 2 = 60
ω∈Λ∗
ω
4
and g 3 = 140
ω∈Λ∗
ω
6
so that (P
′ )
2 = 4P
3 − g 2 P − g 3. Suppose Ω is a simply connected domain
that excludes the three roots of the polynomial 4z
3 − g 2 z − g 3. For w 0 ∈ Ω
and w 0 fixed, define the function I(w) on Ω by
I(w) =
∫ (^) w
w 0
dz √ 4 z^3 − g 2 z − g 3
for w ∈ Ω.
Show that the function z = I(w) has an inverse given by w = P(z + α)
for some constant α; that is, I (P(z + α)) = z for an appropriate α. [Hint:
Prove that (I (P(z + α)))
′ = ±1 and use the fact that P is even.]
