Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 303–314

INFINITELY MANY ASYMPTOTIC VALUES OF LOCALLY UNIVALENT MEROMORPHIC FUNCTIONS

Karl F. Barth and Philip J. Rippon

Syracuse University, Department of Mathematics Syracuse, NY 13244, U.S.A.; kfbarth@mailbox.syr.edu The Open University, Department of Pure Mathematics Walton Hall, Milton Keynes MK7 6AA, United Kingdom; p.j.rippon@open.ac.uk

Abstract. McMillan and Pommerenke showed that a locally univalent meromorphic function f in the disc, without Koebe arcs, has at least three asymptotic values in each boundary arc. The modular function shows that the number three is best possible. We show that if f satisfies certain further conditions, each of which narrowly excludes the modular function, then the number of asymptotic values in each boundary arc must be infinite.

1. Introduction

Let D denote the unit disc {z : |z| < 1}, C the unit circle {z : |z| = 1}, and 𝒞̂ the extended complex plane. Let the function f be meromorphic in D. We say that f is locally univalent if it has at most simple poles and f′ ≠ 0, and that f has Koebe arcs if there are curves J_{n} ⊂ D such that, for some α < β < α + 2π and some a ∈ 𝒞̂,

(a) J_{n} meets the radii arg z = α and arg z = β, for n = 1, 2, …;

(b) |z| → 1, for z ∈ J_{n}, n → ∞;

(c) f(z) → a, for z ∈ J_{n}, n → ∞. A curve γ : z(t), 0 ≤ t < 1, in D is a boundary path if |z(t)| → 1 as t → 1. The set \overline{γ} ∩ C is called the end of γ. We say that f has the asymptotic value a ∈ 𝒞̂ if there is a boundary path γ : z(t), 0 ≤ t < 1, such that

f(z(t)) → a as t → 1.

If f maps γ one-to-one onto a line segment, then the asymptotic value a is said to be linearly accessible along γ. Whenever the end of γ is contained in a subset E of C, we say that f has the asymptotic value a in E; if the end of γ is a singleton {ζ}, then we say that f has the (point) asymptotic value a at ζ. It is clear that if f has no Koebe arcs, then all the asymptotic values of f are point asymptotic values.

In [17], McMillan and Pommerenke proved the following elegant result.

2000 Mathematics Subject Classification: Primary 30D40, 32A18.

Theorem A. If f is meromorphic and locally univalent in D, and has no Koebe arcs, then f has at least three distinct point asymptotic values in each non-trivial arc of C.

The modular function is locally univalent and its only asymptotic values are 0, 1 and ∞; see [14, p. 56] for a summary of facts about the modular function. Thus the number three is best possible in Theorem A. In this paper we show that if f satisfies certain further conditions, each of which narrowly excludes the modular function, then Γ_{P} (f, γ), which denotes the set of point asymptotic values of f in any non-trivial arc γ of C, is infinite.

First we give a result for locally univalent analytic functions for which the maximum modulus M(r, f) = max{|f(z)| : |z| = r} does not grow too quickly.

Theorem 1. Let f be analytic and locally univalent in D, with

(1.1)  (1 − r) log^{+} M(r, f) = o(1)  as r → 1.

Then for each non-trivial arc γ in C, the set Γ_{P} (f, γ) is infinite.

The modular function satisfies (1 − r) log^{+} M(r, f) = O(1), because it is normal; see below. Thus the condition o(1) in (1.1) cannot be replaced by O(1). Also, the assumption that f is locally univalent cannot be dropped. Indeed, there are unbounded analytic functions in D which grow ‘arbitrarily slowly’ but have just one asymptotic value, namely ∞; see [13] and the references therein.

We give an example in Section 4 to show that the conclusion of Theorem 1 cannot be strengthened to ‘uncountable’ no matter what restriction is placed on the growth of an unbounded M(r, f). We remark that Bagemihl [3] gave several conditions on unbounded analytic functions f in D under which the set Γ_{P} (f, γ) is always of positive linear measure.

Next recall that a meromorphic function f is said to be normal if the functions

f(ϕ(z)), where ϕ(z) = e^{iθ}(\frac{z + a}{1 + āz} ), |a| < 1, θ ∈ ℛ,

form a normal family or, equivalently, if

(1.2)  sup_{z∈D}(1 − |z|^{2})f^{#}(z) < ∞,  where f^{#}(z) = \frac{|f′(z)|}{1 + |f(z)|^{2}} ;

see [10]. For example, the modular function is normal because it omits the three values 0, 1 and ∞. By a theorem of Bagemihl and Seidel [4], normal meromorphic functions do not have Koebe arcs, so all their asymptotic values are point asymptotic values, and even angular limits by a theorem of Lehto and Virtanen [12]. Thus Theorem A applies to normal meromorphic functions that are locally univalent, and the modular function again shows that the number three is best possible.

Note that there exist normal meromorphic functions in D with no asymptotic values. See [12] for an example based on a modification of the modular function; alternatively, certain other Schwarz triangle functions with multiple points give an example directly (those described in [18, p. 294, case 3]).

The class 𝒩_{0} consists of functions meromorphic in D such that

(1.3)  (1 − |z|^{2})f^{#}(z) → 0  as |z| → 1.

Such ‘little normal’ functions have been characterised in various ways; see [2], and also [7] where they were called ‘strongly normal’. As noted in [1, p. 31], the hypothesis (1.3) means that the spherical radius of the largest schlicht disc around f(z) on the Riemann image surface of f tends to 0 as |z| → 1. In particular, every univalent function is in 𝒩_{0} . The ‘little Bloch’ class ℬ_{0} consists of analytic functions in D which satisfy

(1 − |z|^{2})|f′(z)| → 0 as |z| → 1,

and these evidently lie in 𝒩_{0} . There are functions in ℬ_{0} which have finite angular limits almost nowhere on C, but all such functions must have finite angular limits on a set of Hausdorff dimension 1, by a result of Makarov; see [20, Chapters 8 and 11].

For locally univalent functions in 𝒩_{0} , the conclusion of Theorem A can be greatly strengthened.

Theorem 2. Let f be meromorphic in D, locally univalent and in 𝒩_{0} . Then for each non-trivial arc γ in C, the set Γ_{P} (f, γ) is of positive linear measure.

We remark that the set of asymptotic values of a meromorphic function is known to be analytic and hence linearly measurable; see [16]. The conclusion of Theorem 2 is best possible, and the modular function shows that the hypothesis in Theorem 2 that f ∈ 𝒩_{0} cannot be replaced by the hypothesis that f is normal.

Finally, recall that a function f analytic in D is said to be in the MacLane class 𝒜 if f is non-constant and has point asymptotic values at a dense set of points in C; see [14]. It is known that if

(1.4)  ∫_{0}^{1} log^{+} log^{+} M(r, f) dr < ∞,

then f ∈ 𝒜 ; see [11]. In particular, normal analytic functions are in 𝒜 because they satisfy (1 − r) log^{+} M(r, f) = O(1) as r → 1; see [10, p. 165].

The class 𝒜_{m}, introduced in [5], is the extension of the class 𝒜 to meromorphic functions. Theorem A shows that locally univalent normal meromorphic functions belong to 𝒜_{m}. Theorem 2 shows that a stronger result holds if ‘normal’ is replaced by 𝒩_{0} . We say that a set is uncountably dense in C if the set meets each non-trivial arc of C in an uncountable set.

Corollary 1. Let f be meromorphic in D, locally univalent and in 𝒩_{0} . Then f has angular limits at an uncountably dense set of points of C.

It is natural to ask about the asymptotic values of a meromorphic function f in 𝒩_{0} without the hypothesis of local univalence. If

(1 − |z|^{2})f^{#}(z) = O(1)(1 − |z|)^{ε} as |z| → 1,

where ε > 0, then it follows from a result of Carleson [6, p. 61] that f has angular limits at all points of C apart from a set of (an appropriate) capacity zero. It is plausible that if f is meromorphic in D and in 𝒩_{0} , then f belongs to 𝒜_{m}; we intend to return to this matter in another paper.

2. Proof of Theorem 1

It is sufficient to consider the case when the function f is unbounded near each interior point of the arc γ. Indeed, if f is bounded near some interior point ζ of γ, then Γ_{P} (f, γ) has positive linear measure by [8, p. 120].

We assume that the set Γ_{P} (f, γ) \ {∞} of finite asymptotic values of f at points of γ is finite, and obtain a contradiction. It follows from (1.1), (1.4) and Cauchy’s estimate that f and f′ belong to the class 𝒜 . Since f′ ≠ 0, we deduce from [15, Theorem 9] that f has at least three distinct point asymptotic values in each non-trivial arc of C. Hence there is a cross-cut γ′ of D ending at distinct points of γ on which f is bounded. Let D(γ) be the component of D \ γ′ such that ∂D(γ) ∩ C ⊂ γ. We can choose R > 0 so that

(2.1)  Γ_{P} (f, γ) \ {∞} ⊂ {|w| ≤ R},

and

(2.2)  |f(z)| ≤ R,  for z ∈ γ′.

Since f is unbounded near each interior point of γ, there is at least one component, Ω say, of {z : |f(z)| > R + 1} lying entirely in D(γ), by (2.2). Pick z_{0} ∈ Ω, let w_{0} = f(z_{0}), and let g denote the branch of f^{−1} such that g(w_{0}) = z_{0} . By (2.1), (2.2) and the local univalence of f, the branch g can be analytically continued along any path from w_{0} in {|w| > R} without meeting any singularity. Indeed, since the Riemann surface of f has no branch points, such a singularity would give rise to an asymptotic value α of f, with |α| > R, at a point of γ.

Thus if H = {t : Re(t) > log R} and t_{0} satisfies e^{t_{0}} = w_{0} , then h(t) = g(e^{t}) can be analytically continued from t_{0} along any path in H to give a single-valued analytic function, by the monodromy theorem. Two cases can then arise [18, p. 283], as follows.

(a) The function h is univalent in H.

(b) The function h is periodic with period 2πim, where m is a minimal positive integer.

In the present situation case (b) cannot occur. Indeed, if case (b) holds, then the function G(w) = g(w^{m}) must be univalent in Δ_{0} = {|w| > R^{1/m}}, so it has a Laurent expansion there of the form

G(w) = a_{1}w + a_{0} + a_{−1}w^{−1} + · · · .

Since g takes values in D, we have a_{1} = 0. Hence G has a univalent extension to Δ_{0} ∪ {∞} and so a_{−1} ≠ 0. Then G(∞) = a_{0} is a pole of f in D, which is impossible.

Thus only case (a) can occur. Since the values taken by the function h in H all lie in the component of {z : |f(z)| > R} which contains Ω, we deduce that Ω is conformally equivalent to {t : Re(t) > log(R + 1)} and that ∂Ω consists of a simple analytic curve in D(γ).

Now we use the fact that f belongs to the MacLane class 𝒜 . Then by [14, Theorem 1], both ends of the simple curve ∂Ω must approach points of γ. Moreover, by the Carathéodory boundary correspondence theorem [20, p. 24], both ends of ∂Ω must approach the same point, ζ_{0} say, of γ. It follows that the function

u(z) = \{ \begin{matrix}log(|f(z)|/(R + 1)), & for z ∈ Ω, \\ 0, & for z ∈ D \ Ω, \end{matrix}

is subharmonic in D and tends to zero at each point of C \ {ζ_{0}}. The hypothesis (1.1) implies that max{u(z) : |z| = r} = o(1)/(1 − r) as r → 1 so, by a special case of Dahlberg’s radial maximum theorem [9, Theorem 2], u must be identically zero, a contradiction. This completes the proof of Theorem 1.

Remark. By a result of MacLane [15, Theorem 8], it follows that under the hypotheses of Theorem 1 the set of linearly accessible finite asymptotic values of f is unbounded and so infinite.

3. Proof of Theorem 2

First we consider the case when there exists α ∈ 𝒞̂ such that f is bounded away from α, in the spherical metric, near some interior point ζ of γ. Then f or g(z) = 1/(f(z) − α) is bounded near ζ, and it follows from [8, p. 120] that Γ_{P} (f, γ) is of positive linear measure.

There remains the case when, for each α ∈ 𝒞̂, the function f takes values arbitrarily close to α, in the spherical metric, in any neighbourhood of any interior point of γ. To deal with this case, we suppose that the set Γ_{P} (f, γ) is of linear measure zero, and obtain a contradiction.

Since normal functions do not have Koebe arcs [4], we deduce from Theorem A that there is a cross-cut γ′ of D ending at distinct points of γ on which f is bounded, say

(3.1)  |f(z)| ≤ R,  for z ∈ γ′,

for some R > 3. Let D(γ) be the component of D \ γ′ such that ∂D(γ) ∩ C ⊂ γ. Since f takes values arbitrarily close to any α ∈ 𝒞̂ near each point of γ, there is some z_{0} ∈ D(γ) such that

(3.2)  R + 2 < |f(z_{0})| < 2R,

and, by (1.3),

(3.3)  (1 − |z_{0}|^{2})f^{#}(z_{0}) < \frac{1}{1 + 4R^{2}} .

Moreover, since f is an open mapping and Γ_{P} (f, γ) has linear measure zero, we can assume, by varying z_{0} slightly if necessary, that

(3.4)  w_{0} = f(z_{0}) does not belong to Γ_{P} (f, γ).

From (3.2) and (3.3), we deduce that

(3.5)  (1 − |z_{0}|^{2})|f′(z_{0})| < 1.

For z ∈ D, let d(z) denote the radius of the maximal schlicht disc in the image surface ℱ of f centred at the point of ℱ corresponding to f(z). Since f is locally univalent the surface ℱ has no branch points, so this maximal schlicht disc has a (linearly accessible) asymptotic value of f on its boundary corresponding to an inverse function singularity of f. It follows from Schwarz’s lemma (see [21]) that if f is analytic at z, then

(3.6)  d(z) ≤ (1 − |z|^{2})|f′(z)|.

We claim that there is a path γ_{0} in D with one endpoint at z_{0} such that

(3.7)  there exist distinct ζ_{n} ∈ γ_{0} such that f(ζ_{n}) = w_{0}, for n = 1, 2, …,

and

(3.8)  |f(z) − w_{0}| ≤ 2d(z_{0}),  for z ∈ γ_{0}.

To prove this claim, let Σ_{0} denote the maximal schlicht disc in ℱ centred at the point of ℱ corresponding to w_{0} = f(z_{0}) and let w′_{0} be a singularity of f^{−1} on the boundary of Σ_{0} . Then |w′_{0} − w_{0}| = d(z_{0}) < 1 by (3.5) and (3.6). Also, let Γ′_{0} denote the radius of Σ_{0} from w_{0} to w′_{0} .

Next let g denote the branch of f^{−1} such that g(w_{0}) = z_{0} , so g is univalent in Σ_{0} . Let Σ′_{0} denote the open disc in 𝒞 with diameter along the line segment Γ′_{0} .

Then Σ′_{0} has radius \frac{1}{2}d(z_{0}) and centre \frac{1}{2}(w_{0} + w′_{0}) = w_{0}^{∗}, say. We consider the analytic continuation of g from near w_{0} along paths in 𝒞 \ \overline{Σ′_{0}} .

First suppose that such analytic continuation of g meets no singularities of f^{−1} . Choose t_{0} such that

(3.9)  w_{0}^{∗} + \frac{1}{2}d(z_{0})e^{t_{0}} = w_{0}.

Then the function

h(t) = g(w_{0}^{∗} + \frac{1}{2}d(z_{0})e^{t}),

can be analytically continued along any path in H = {Re(t) > 0}, starting near t_{0} . Thus h is a single-valued analytic function in H, by the monodromy theorem, and h(t_{0}) = z_{0} . Two cases can then arise [18, p. 283], as follows.

(a) The function h is univalent in H.

(b) The function h is periodic with period 2πim, where m is a minimal positive integer. Once again, case (b) cannot occur in the present situation. Indeed, if case (b)

holds, then the function G(w) = g(w_{0}^{∗} + (w − w_{0}^{∗})^{m}) must be univalent in Δ_{0} =

\{|w − w_{0}^{∗}| > (\frac{1}{2}d(z_{0}))^{1/m}\}, so it has a Laurent expansion there of the form

G(w) = a_{1}(w − w_{0}^{∗}) + a_{0} + a_{−1}(w − w_{0}^{∗})^{−1} + · · · .

Since g takes values in D, we have a_{1} = 0. Hence G has a univalent extension to Δ_{0} ∪ {∞} and so a_{−1} ≠ 0. Then G(∞) = a_{0} is a pole of f which must be simple. Writing z = G(w), we obtain f(z) = w_{0}^{∗} + (w − w_{0}^{∗})^{m} and

z − a_{0} = \frac{a_{−1}}{w − w_{0}^{∗}} (1 + o(1)) as w → ∞,

so

f(z) = w_{0}^{∗} + \frac{(a_{−1})^{m}}{(z − a_{0})^{m}} (1 + o(1)) as z → a_{0}.

Thus m = 1, so g is univalent in Δ_{0} ∪ {∞}. Because ∂Δ_{0} \ {w′_{0}} ⊂ Σ_{0} , it follows that g is analytic in Σ_{0} ∪ (Δ_{0} ∪ {∞}) = 𝒞̂\ {w′_{0}}. Since g takes values in D, this contradicts Liouville’s theorem.

Thus case (a) occurs, so h is univalent in H. Moreover, we can find open discs D_{n} with centres t_{n} = t_{0} + 2πin, n ∈ 𝒵 , such that h is univalent in the larger set

H′ = H ∪ ⋃_{n∈𝒵} D_{n}.

This follows from (3.9) and the fact that the analytic continuation of g along any path in 𝒞 \ \overline{Σ′_{0}} which ends at w_{0} can be extended to a neighbourhood of w_{0} ; for otherwise w_{0} ∈ Γ_{P} (f, γ) contrary to (3.4).

We now take a simple path β_{0} in H′ ∩ {Re(t) < 1} with one endpoint at t_{0} , passing through every t_{n}, n = 0, 1, 2, …, and tending to ∞. Then γ_{0} = h(β_{0}) is a simple path in D with one endpoint at z_{0} . Since f(h(t)) = w_{0}^{∗} + \frac{1}{2}d(z_{0})e^{t}, we deduce that (3.7) holds, with ζ_{n} = h(t_{n}), and also (3.8) holds, as required.

Suppose, on the other hand, that the analytic continuation of g from w_{0} along some path Γ′′_{0} in 𝒞 \ \overline{Σ′_{0}} does encounter a singularity of f^{−1} , at w′′_{0} say. Choose r_{0} > 0 such that r_{0} < min{d(z_{0}), dist (w′_{0}, Γ′′_{0})}. Then let Γ_{0} be a Jordan curve in 𝒞 which surrounds w′_{0} and consists of two chords through w_{0} of the disc \overline{Σ′_{0}} together with an arc of a circle \{|w − w′_{0}| = r\}, where 0 < r < r_{0} . Since Γ_{P} (f, γ) has linear measure zero, it fails to meet almost all such chords and almost all such circular arcs, so we can assume by (3.2) that Γ_{0} ⊂ {|w| > R} \ Γ_{P} (f, γ). In particular, the branch g can be analytically continued from w_{0} around Γ_{0} arbitrarily far (in either direction). Note that by construction Γ_{0} , Γ′_{0} , Γ′′_{0} meet only at w_{0} .

Let γ_{0} , γ′_{0} , γ′′_{0} denote the paths in D obtained by analytic continuation of g along Γ_{0} , Γ′_{0} , Γ′′_{0} , respectively. Then γ′_{0} and γ′′_{0} are asymptotic paths meeting only at z_{0} , so the path γ′_{0} ∪ γ′′_{0} forms a cross-cut of D. The path γ_{0} can meet γ′_{0} ∪ γ′′_{0} only at z_{0} . Thus, by the local univalence of f, the path γ_{0} must pass through z_{0} exactly once, as it crosses γ′_{0} ∪ γ′′_{0} . Also, if the part of γ_{0} on one side of the cross-cut were to intersect itself, then at the first point of intersection there would again be a contradiction to local univalence. It follows that γ_{0} is a simple path passing through z_{0} such that (3.7) and (3.8) hold. Thus we have established the claim.

It follows from (3.7) that |ζ_{n}| → 1 as n → ∞. Hence d(ζ_{n}) → 0 as n → ∞ by (1.3) and (3.6), because f is bounded on γ_{0} by (3.8). Thus we can truncate γ_{0} at a point, z_{1} say, chosen from the ζ_{n}, such that

f(z_{1}) = w_{0}, 1 − |z_{1}| < \frac{1}{2} and d(z_{1}) ≤ \frac{1}{2}d(z_{0}) < \frac{1}{2}. Now we repeat the above argument with z_{0} replaced by z_{1} to obtain a path

γ_{1} in D with one endpoint at z_{1} such that

there exist distinct ζ_{n} ∈ γ_{1} such that f(ζ_{n}) = w_{0}, for n = 1, 2, …, and

|f(z) − w_{0}| ≤ 2d(z_{1}), for z ∈ γ_{1}.

We then truncate γ_{1} at a point z_{2} such that

f(z_{2}) = w_{0}, 1 − |z_{2}| < \frac{1}{4} and d(z_{2}) ≤ \frac{1}{2}d(z_{1}) < \frac{1}{4}.

By repeating this process, we obtain a sequence of such truncated paths γ_{n}, n = 0, 1, 2, …, such that f(z) tends to w_{0} along the path

γ_{∞} = γ_{0} ∪ γ_{1} ∪ γ_{2} ∪ ⋯,

and, by (3.2), |f(z)| > R for z ∈ γ_{∞}. By (3.1) the end of γ_{∞} must lie in γ, so γ_{∞} must tend to a point of γ because f has no Koebe arcs. Hence w_{0} ∈ Γ_{P} (f, γ), which contradicts (3.4). The proof of Theorem 2 is complete.

Remarks. 1. The proof can be adapted to show that if, in any neighbourhood of any point of γ, the locally univalent function f in 𝒩_{0} takes values arbitrarily close to each α ∈ 𝒞̂, in the spherical metric, then Γ_{P} (f, γ) has positive linear measure in any neighbourhood of each point of 𝒞̂. This holds near ∞ by taking R arbitrarily large in (3.1) and, for other points α, we consider the function in 𝒩_{0} obtained by composing f with a rotation of the Riemann sphere taking α to ∞. It seems possible that under these circumstances we must actually have Γ_{P} (f, γ) = 𝒞̂.

2. An earlier version of this proof, which was more like the proof of Theorem 1, examined the components of {z : |f(z)| > R} in D(γ). This raised the question of whether the level sets L(R) = {z : |f(z)| = R}, R > 0, end at points of C, in the sense of MacLane [14, p. 8]. In fact, the proof of Theorem A in [17] shows that the ‘three asymptotic values’ obtained in each arc are linearly accessible. It follows that if f satisfies the hypotheses of Theorem A, then every level set L(R), R > 0, of f must end at points.

4. An example

We show here that the conclusion in Theorem 1 cannot be strengthened to ‘uncountable’.

Example 1. Let μ be an increasing function on [0, 1) such that μ(0) = 1 and μ(r) → ∞ as r → 1. Then there exists a function f analytic and locally univalent in D, such that

(4.1)  M(r, f) ≤ μ(r),  for 0 ≤ r < 1,

and the set of asymptotic values of f is countable.

Our approach is to follow the geometric construction of MacLane [13] for an analytic function in D of ‘arbitrarily slow growth’ with level curves C_{n}, n = 1, 2, …, which are nested Jordan curves that expand towards C, on which |f| = ϱ_{n}, n = 1, 2, …, where ϱ_{n} → ∞ as n → ∞. The method is to construct the image Riemann surface of the function by a process of successive enlargements, and the main difference in our construction is that we replace MacLane’s adjoined two-sheeted surfaces by adjoined logarithmic spirals.

We first fix a positive strictly increasing sequence ϱ_{n}, with ϱ_{n} → ∞ as n → ∞, and put α_{n} = n\sqrt{2} , n = 1, 2, …. Then define

(4.2) a_{n,k} = ϱ_{n} exp(iθ_{n,k}),  where θ_{n,k} = 2πα_{n} + \frac{2πk}{ν_{n}} , k = 1, 2, …, ν_{n}.

For n = 1, 2, …, the points a_{n,k}, k = 1, 2, …, ν_{n}, are equally spaced on the circle {|w| = ϱ_{n}} and no two a_{n,k}, k = 1, 2, …, ν_{n}, n = 1, 2, …, are in the same direction from 0. The sequences ϱ_{n} and α_{n} are fixed throughout, but ν_{n} will be varied to determine the required Riemann surface.

Let

L(n, k) = {|w| ≥ ϱ_{n}, arg w = θ_{n,k}}, k = 1, 2, …, ν_{n}, n = 1, 2, …,

so the rays L(n, k), k = 1, 2, …, ν_{n}, n = 1, 2, …, are mutually disjoint. The required surface 𝒮 is built in stages. We begin with the surface 𝒮_{1} consisting of the w-plane slit along all rays L(1, k), k = 1, 2, …, ν_{1} . Then the surface 𝒮_{2} is constructed by first adjoining a half-logarithmic spiral to each edge of each slit in 𝒮_{1} , and then making a slit over every ray L(2, k), k = 1, 2, …, ν_{2} , in all the resulting sheets. The surface 𝒮_{3} is constructed in an analogous manner by adjoining half-logarithmic spirals to edges of existing slits in 𝒮_{2} wherever possible and then making new slits over the rays L(3, k), k = 1, 2, …, ν_{3} , wherever possible.

Repeating this process indefinitely, we obtain a surface 𝒮 without algebraic branch points. The surface 𝒮 is simply connected because at each stage the addition of each pair of half logarithmic spirals preserves the property of being simply connected, and 𝒮 = ⋃_{n=1}^{∞} 𝒮_{n}. Now let 𝒯_{n} denote that component of 𝒮_{n} over {|w| < ϱ_{n}} which contains the origin in 𝒮_{1} . If ν_{1}, ν_{2}, …, ν_{n−1} are fixed and ν = ν_{n} is variable, then as ν → ∞ the sequence of surfaces 𝒯_{n+1} = 𝒯_{n+1}(ν) converges to its Carathéodory kernel, the unique maximal Riemann surface all of whose compact subsets K can be embedded in 𝒯_{n+1}(ν), for ν ≥ ν(K). This kernel is 𝒯_{n}. We now need the following generalisation of the Carathéodory kernel theorem, due essentially to L. I. Volkovyskii; see [13].

Theorem B. Let R_{n}, n = 1, 2, …, be a sequence of positive numbers such that lim_{n→∞} R_{n} = R, 0 < R < ∞. Let ℱ_{n} = {|z| < R_{n}} and ℱ = {|z| < R}. Let [𝒢_{n}; Q_{n}] be a sequence of Riemann surfaces over the w-plane, each containing a schlicht disc of radius s_{0} > 0 about Q_{n}, where Q_{n} is a point of 𝒢_{n} over w = 0. Let w = F_{n}(z) be holomorphic in ℱ_{n} and map ℱ_{n} one-one onto 𝒢_{n} with F_{n}(0) = Q_{n} and F′_{n}(0) = 1. Let Φ_{n} be the inverse of F_{n}. Then the following are true.

(a) If F_{n} → F locally uniformly on ℱ, then [𝒢_{n}; Q_{n}] converges to its kernel [𝒢; Q] and F maps ℱ one-one onto 𝒢 , with F (0) = Q and F′(0) = 1. Also, Φ_{n} → Φ locally uniformly on 𝒢 and Φ is the inverse of F.

(b) If {F_{n}} is a normal family on ℱ and if [𝒢_{n}; Q_{n}] converges to its kernel [𝒢; Q], then F_{n} → F locally uniformly on ℱ.

Now let f_{n} map {|z| < r_{n}} onto 𝒯_{n}, with f_{n}(0) = 0 ∈ 𝒮 and f′_{n}(0) = 1. Since g_{n} = f_{n+1}^{−1} ∘ f_{n} maps {|z| < r_{n}} into {|z| < r_{n+1}} with g_{n}(0) = 0 and g′_{n}(0) = 1, we deduce that

(4.3)  r_{n} < r_{n+1},  for n = 1, 2, ….

Let δ_{n} be any positive sequence such that

(4.4)  ∑_{n=1}^{∞} δ_{n} < ∞.

We shall show that the sequence ν_{n}, n = 1, 2, …, can be chosen so that

(4.5)  r_{n+1} < r_{n} + δ_{n},  for n = 1, 2, ….

To do this, suppose that ν_{1}, ν_{2}, …, ν_{n−1} have already been chosen. As above, we indicate dependence on ν = ν_{n} by writing 𝒯_{n+1} = 𝒯_{n+1}(ν), f_{n+1}(z) = f_{n+1}(z; ν) and r_{n+1} = r_{n+1}(ν). Now choose a subsequence of values of ν so that

r_{n+1}(ν) → r_{n+1}(∞) ≥ r_{n}.

We claim that r_{n+1}(∞) = r_{n}. Indeed, the subsequence f_{n+1}(z; ν) is defined eventually on each compact subset of {|z| < r_{n+1}(∞)}, is uniformly bounded by ϱ_{n+1} and hence forms a normal family. Also, the sequence 𝒯_{n+1}(ν) converges to its kernel 𝒯_{n}. Thus, by Theorem B, part (b),

f_{n+1}(z; ν) → f_{n+1}(z; ∞) as ν → ∞,

locally uniformly and, by Theorem B, part (a), f_{n+1}(z; ∞) maps {|z| < r_{n+1}(∞)} onto 𝒯_{n} with the same normalization as f_{n}. Hence r_{n+1}(∞) = r_{n}. Thus we can indeed choose ν_{n} so that (4.5) holds.

Now we apply Theorem B to the sequences f_{n} and 𝒯_{n}. By (4.4), r_{n} → R < ∞ and by construction 𝒯_{n} converges to its kernel 𝒮 . To show that the sequence f_{n} is normal, let f^{∗} map {|z| < R^{∗}} = D^{∗} onto 𝒮 with f^{∗}(0) = 0 ∈ 𝒮 and (f^{∗})′(0) = 1; here 0 < R^{∗} ≤ ∞. Since 𝒯_{n} ⊂ 𝒮 , the functions h_{n} = (f^{∗})^{−1} ∘ f_{n} are conformal and normalised in the usual way. Thus the family {h_{n}} is normal and it follows that the family f_{n} = f^{∗} ∘ h_{n} is normal in {|z| < R}. Hence, by Theorem B, part (b), f_{n} → f locally uniformly in {|z| < R}, where f maps {|z| < R} onto 𝒮 , the kernel of the sequence 𝒯_{n}, with f(0) = 0 ∈ 𝒮_{1} and f′(0) = 1.

The proof that the sequence ν_{n} can be chosen in such a way that the condition (4.1) holds, is based on (4.3) and (4.4), as in [13, Section 3]. Finally, it is clear from the construction of 𝒮 that the function f mapping {|z| < R} onto 𝒮 is locally univalent and has countably many finite asymptotic values, namely a_{n,k}, k = 1, 2, …, ν_{n}, n = 1, 2, …. This completes the proof of Example 1.

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Received 14 November 2002