Throughout this section, we assume that Ω is an unbounded simply connected domain in ℂ with 0 ∈ Ω. As in the previous section, *σ* represents the Riemann mapping of onto Ω with *σ*(0) = 0 and *σ*'(0) > 0. We will also define *φ* via the Schröder functional equation. Throughout we let

so that *δ*_{Ω}(*w*) is the Euclidean distance from *w* to the boundary of Ω.

**Theorem 1**. *Let φ be a univalent function mapping**into*, *φ*(0) = 0. *Suppose that the closure of**intersects**only at finitely many fixed points and is contained in a Stolz angle of opening no greater than απ there*.

*If* |*φ*'(0)| > 16 tan(*απ/* 2) *then the following are equivalent*

*1*. *C*_{
φ
} *is compact on*;

*2*. ;

*3*. *For every n* > 0, .

Remark: It has recently been shown by Smith [6] that compactness of *C*_{
φ
} on is equivalent to compactness of *C*_{
φ
} on , *BMOA* and *VMOA* when *φ* is univalent and so in the above theorem, the first condition could read: *C*_{
φ
} is compact on , , *BMOA* and *VMOA* Before proceeding, we prove the following lemma.

**Lemma 1**. *Under the hypotheses of the theorem, w and γw tend to the same prime end at* ∞, *and ∂γ* Ω ⊂ Ω.

*Proof*. The first assertion follows from the fact that the closure of touches only at fixed points. Suppose now that the second assertion is false and there are distinct prime ends *ρ*_{1} and *ρ*_{2} with *ρ*_{1} = *γρ*_{2}. Then under the boundary correspondence given by *σ* there are distinct points *η*, with

It follows that and therefore *ζ* is a fixed point of *φ*. Hence, we have the contradiction *ρ*_{1} = *ρ*_{2}. □

*Proof*. We first prove that 1 is equivalent to 2.

By the results of Madigan and Matheson [2], and Smith [6] cited above *C*_{
φ
} is compact on if and only if

However, by Schröder's equation

Since Ω is simply connected, *λ*_{Ω} (*w*) ≍ 1/*δ*_{Ω} (*w*) and so *C*_{
φ
} is compact on if and only if

Since *γ* Ω ⊂ Ω, *γw* → ∂Ω implies that *w →* ∂Ω. Therefore, (3) holds if and only if

By the Lemma, we see that *γw* → ∂Ω means *w* → ∞ and *w* ∈ *γ* Ω, and we have shown that 1 and 2 are equivalent.

Suppose that 2 holds and let *ε* > 0 be given. Then we can find a *R* > 0 so that *δ*_{Ω} (*w*) *<εδ*_{Ω} (*γw*) for all |*w*| > *R*, since there are only a finite number of prime ends at *∞*. Choose *w* ∈ Ω arbitrarily with modulus greater than *R* and let *n* satisfy |*γ*|^{-n}*R* < |*w*| ≤ |*γ* |^{-n -1}*R*.

Then we have that *δ*_{Ω} (*w*) *< ε*^{n}*δ*_{Ω} (*γ*^{n} *w*) and hence

Now as *w* → ∞ in *γ* Ω, *γ*^{n}*w* lies in a closed set properly contained in Ω and therefore *δ*_{Ω} (*γ*^{n} *w*) is bounded below by a constant independent of *w*. We thus have that

and since *ε* was arbitrary, the left-hand side of the above inequality must tend to ∞. Hence, we have shown that lim_{w→∞}|*w*| ^{β} *δ*_{Ω} (*w*) = 0 for every *β* > 0.

Now may be interpreted geometrically as lim_{w→∂Ω}*n*|*w*| ^{n-1}*δ*_{Ω} (*w*) = 0 and this follows from the above argument. Therefore, 2 implies 3.

To show that 3 implies 2, we need to show that if

then 2 holds.

To complete the proof, we require the following lemma whose proof we merely sketch.

**Lemma 2**. *Under the hypotheses of the theorem*,

*Sketch of Proof*. First note that

Now if lies in a non-tangential angle of opening *απ* at *ζ*, then a short calculation shows that

and the assertion follows. □

Now with *f* defined above, we have

for large enough *w*. Hence,

as *w* → ∞ and so 2 holds. □

It is of interest to consider the growth of *σ* since condition 3 would imply that it has very slow growth. The following corollary follows from 3 and the fact that functions in grow at most of order log 1/(1 *-* |*z*|).

**Corollary 1**. *Suppose that φ satisfies the hypotheses of the Theorem and that any of the equivalent conditions holds, then for r* = |*z*|.

We also provide the following restatement of the hypotheses of Theorem 1 to illustrate the main properties of the Königs domain.

**Corollary 2**. *Let* Ω *be an unbounded domain in ℂ with γ* Ω ⊂ Ω *and* 0 ∈ Ω*. Suppose that has* Ω *only finitely many prime ends at* ∞ *and*

*In addition, suppose that* ∂*γ* Ω ⊂ Ω. *If*, *σ*(0) = 0, *σ*'(0) > 0, *and φ is defined by Schröder's equation, then the following are equivalent*.

*1*. *C*_{
φ
} *is compact on*;

*2*. ;

*3*. *For every n* > 0, .

The hypothesis on the boundary of Ω is vital. If we do not assume that ∂*γ* Ω ⊂ Ω, then we deduce from the proof of the Theorem that is equivalent to

In this situation, the finite part of the boundary of Ω plays a complicated role in the behaviour of *φ*. We conclude this section by constructing a domain that displays very bad boundary properties. This answers a question of Madigan and Matheson in [2].

In [2] it was shown that if ∂*φ*(*D*) touches in a cusp, then . However, it is not sufficient that ∂*φ*(*D*) touches at an angle greater that 0. The question was raised of whether or not it is possible that can be infinite.

With the hypothesis that ∂*γ* Ω ⊂ Ω the prime ends at ∞ correspond to points of that touch . Therefore, is at most countable. A natural question to ask is whether or not can ever be positive, where Λ represents linear measure.

This example is well known in the setting of the unit disk, see [7, Corollary 5.3]. We describe here the construction in terms of the Königs domain.

**Theorem 2**. *There is a univalent function**such that*.

*Proof*. We construct the domain Ω so that it satisfies (4). Let 0 < *γ* < 1 be given. We will define a nested sequence , *n* = 1, 2, ... so that

where Θ _{
n
} ⊂ Θ_{n+1}for all *n* = 1, 2, ....

First let *N* > 2 be chosen arbitrarily and let Θ_{1} = {2*πk*/*N* : *k* = 0, ..., *N -* 1}.

Suppose now that Θ _{
n
} has been defined, then let Θ_{n+1}be such that Θ _{
n
} ⊂ Θ_{n+1}and whenever *θ* ∈ Θ _{
n
} is isolated, we define a sequence *θ*_{
k
} ∈ Θ_{n+1}, *k* = 1, 2, ..., so that *θ*_{
k
} → *θ* as *k* → ∞ and for each *k* there is a *j* so that *θ - θ*_{
k
} = *θ*_{
j -θ
} . Moreover, assume that

In this way, we define the sequence of sets Θ_{
n
}, *n* = 1, 2, .... We will, furthermore, assume that for each , there is a sequence *θ*_{
n
} ∈ Θ_{
n
}, *n* = 1, 2, ...., such that *θ*_{
n
} → *θ*.

We claim that this gives the desired domain Ω with boundary defined by (5).

To see this, let *γw* ∈ Ω be arbitrary, then by construction, we may find a *ζ* ∈ ∂Ω so that *δ*_{Ω} (*γw*) = |*ζ - γw*|. It is readily seen that for such *ζ*, there is an *n* so that *ζ* ∈ {*re*^{iθ}: *r* ≥ *γ*^{-n}} for some *θ* ∈ Θ _{
n
} and moreover, *θ* is isolated in Θ_{
n
}.

If we now consider *w*, we may find a sequence *θ*_{
k
} → *θ* as *k* → ∞ so that for all *k* hence we may fix a *k* so that *δ*_{Ω} (*w*) = |*w* - *n*| for .

By estimating the line segment [*w*, *η*] by the arc of joining *w* to *η*, we see that *δ*_{Ω} (*w*) ≍ |*w*|| *α - θ*_{
k
} | where *w* = *re*^{iα}.Therefore, we have the estimate *δ*_{Ω} (*w*) ≤ |*w*||*θ*_{k+1}- *θ*_{
k
} |. By a similar argument, we deduce the estimate *δ*_{Ω} (*γw*) ≍ |*γw*||*θ - θ*_{
k
} | and so

by (6) and so the construction is complete.

We claim that if is defined as usual and *φ* is given by Schröder's equation, then .

In fact, if *θ* ∈ Θ _{
n
} is isolated, then the ray *R* = {*re*^{iθ}: *r* ≥ *γ*^{-n-1}} is contained in a single prime end of Ω. Therefore, to each such ray, there exists a point that corresponds to *R* under *σ*. Since *γR* ⊂ ∂Ω, we thus have that *ζ* corresponds to a prime end *p* under *φ* with .

On the other hand, if *θ* ∈ Θ _{
n
} is isolated, then *R*' = {*re*^{iθ}: *γ*^{-n}≤ *r* < *γ*^{-n-1}} satisfies *γR*'∩ ∂Ω = ∅, and so there is an arc such that *σ* (*ρ*_{
θ
} ) = *R*' and *ρ*_{
θ
} has an end-point in .

Hence, each is contained in a prime end of and

The result follows. □