Second order linear differential equations with a basis of solutions having only real zeros

Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w''+Aw=0$ has two linearly independent solutions with only real zeros, then the order of $A$ must be an odd integer or one half of an odd integer. Moreover, $A$ has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.


Introduction and results
For any entire function A, all solutions of the differential equation (1.1) w + Aw = 0 are entire.We consider the question of when this equation has two linearly independent solutions which have only real zeros.For a polynomial A this is possible only when A is constant [26,Theorem 3].On the other hand, there are transcendental coefficients A for which this happens.However, we will show that if A has finite order, then this is possible only in special cases.In particular, the order must be an odd integer or one half of an odd integer.We begin by making some general remarks on the equation (1.1), all of which can be found in [31].Let w 1 and w 2 be two linearly independent solutions.Then their Wronskian determinant The ratio of two linearly independent solutions F = w 2 /w 1 satisfies the Schwarz equation The function F is meromorphic in C and locally univalent.All meromorphic locally univalent meromorphic functions arise in this way.A normalized pair can be recovered from F by the formulae The product E = w 1 w 2 of a normalized pair of solutions of (1.1) has the property Every entire function satisfying (1.3) is the product of a normalized pair of solutions of (1.1), with A given by Conversely, every entire function E satisfying (1.4) for some entire function A satisfies (1.3).The functions E and F are related by the formula Note that zeros of E correspond to zeros and poles of F .We conclude that studying zeros of linearly independent solutions of (1.1) is essentially equivalent to investigating the zeros of functions satisfying (1.3), or the zeros and poles of locally univalent functions.The relation between the coefficient A in (1.1), the function E satisfying (1.3) and the locally univalent function F is given by (1.2), (1.4) and (1.5).
Functions satisfying (1.3) play an important role in the work of Steven Bank and Ilpo Laine [1,2], and they are now called Bank-Laine functions, or BL functions for short.Since this work, there has been a substantial interest in BL functions of finite order.We refer to the surveys [24] and [32] which cover the literature before 2008, and to the introductions of the recent papers [7,8,36,37].In particular, much attention has been paid to the exponent of convergence λ(E) and the order ρ(E) of a BL function E; see [31, p. 7] for the definitions.
When A is transcendental, all solutions w of (1.1) have infinite order.However, it is possible that the product E of two solutions has finite order.For example, when with a polynomial p, we have a normalized pair of solutions The order of A and of the corresponding BL function E = w 1 w 2 = exp(−2p)/2 is the degree of p and hence an integer.Bank and Laine conjectured that if λ(E) < ∞, then ρ(A) ∈ N, as is the case in this elementary example.Langley [33] constructed non-elementary examples of BL functions and corresponding coefficients A of finite integer order in 1998; further examples were constructed in [13,34].Recently, examples of BL functions of any order in [1, ∞) and corresponding coefficients A of any order in (1/2, ∞) were constructed in [7,8], resolving the conjecture of Bank and Laine in the negative.Note that the order of a transcendental BL function is at least 1; see [50], [48, Corollary 1] and [51, Theorem 1].
In the recent papers [36] and [37], Jim Langley started to investigate real BL functions E of finite order for which all zeros are real.So in this case the associated differential equation (1.1) has two linearly independent solutions with only real zeros.As already mentioned, for a polynomial A this is possible only when A is constant [26,Theorem 3].In fact, if A has degree n, then the exponent of convergence of the non-real zeros of the product of two linearly independent solutions is equal to (n + 2)/2 [23, Theorem 1].So it is surprising that there exist non-elementary BL functions of finite order with only real zeros.
Our first result says that instead of assuming that E has finite order, it is enough to assume that A has finite order.Note that we always have ρ(A) ≤ ρ(E) by (1.4).
Theorem 1.1.Let A and E be entire functions satisfying (1.4).Suppose that the zeros of E lie on finitely many rays emanating from the origin.Then ρ(E) < ∞ if and only if ρ(A) < ∞.
Theorem 1.2 below shows that we actually have ρ(A) = ρ(E) if the zeros of E are real.
Theorem 1.1 yields that studying entire coefficients A of finite order for which the differential equation (1.1) has two linearly independent solutions with only real zeros is equivalent to studying BL functions of finite order with real zeros.
A meromorphic function f is called real if it maps R into R ∪ {∞}.This is equivalent to f (z) = f (z) for all z ∈ C. Functions (not necessarily analytic) which satisfy this last equality will be called symmetric.For other objects, like subsets of the plane, the word symmetric will mean invariant under complex conjugation.
We say that an infinite real sequence without finite accumulation points is one-sided if it is bounded from above or below, and two-sided otherwise.
The results of Langley [36,37] on BL functions with real zeros can be summarized as follows.Recall that λ(E) denotes the exponent of convergence of the zeros of E.
Theorem A. Let E be a real Bank-Laine function of finite order with only real zeros and let A be given by (1.4).
Langley's proofs actually yield a more general result, stated as Theorem B below.He also constructed examples for which we have equality in the estimates of λ(E) in (a) and (b).Of course, if A is constant, then the possible forms of E can be determined explicitly.
We will strengthen Theorem A as follows.
Theorem 1.2.Let E and A be as in Theorem A.
(a) If the zeros of E form an infinite one-sided sequence, then there exists n ∈ N with n ≥ 2 such that λ(E) = ρ(E) = ρ(A) = n − 1/2.(b) If the zeros of E form an infinite two-sided sequence and A is non-constant, then there exists n ∈ N with n ≥ 2 such that λ(E) = ρ(E) = ρ(A) = 2n − 1.
Remark 1.1.For an arbitrary real BL function E with only real zeros, not necessarily of finite order, there are no restrictions on the exponent of convergence of E. Indeed, a result of Shen [49] says that any set without finite accumulation points is the zero set of a BL function.
Theorem 1.2 will be a corollary of a more general result, stated as Theorem 1.3 below.However, we will also give a direct proof of Theorem 1.2 in section 3.This proof is analytic in nature, while the proof of the more general Theorem 1.3 is geometric.The proofs of Theorems 1.2 and 1.3 are independent of each other.
To state Theorem 1.3 and Theorem B, we introduce some terminology.All surfaces in this paper are oriented and have countable base.A continuous map of surfaces F : X → Y is called topologically holomorphic if for every point p ∈ X there are local coordinates at p and at F (p) in which F has the form z → z n , where n is a positive integer.According to Stoïlow [54], all open discrete maps are topologically holomorphic.
The points where n ≥ 2 are called critical points; their images are called critical values.The critical values correspond to the algebraic singularities of the inverse F −1 .The function F is a local homeomorphism if and only if there are no critical points.
The transcendental singularities of the inverse are defined as follows; cf.[5].(There it is assumed that F is meromorphic, but the definition extends to topologically holomorphic functions without change.)Let a ∈ C. Suppose that D → U (D) associates to every topological disk containing a a connected component U (D) of F −1 (D), in such a way that U (D 1 ) ⊂ U (D 2 ) when D 1 ⊂ D 2 .(Note that D → U (D) is determined by its values on any base of neighborhoods of a.) If D U (D) = ∅, then we say that D → U (D) is a transcendental singularity of F −1 over a.In this case, the sets U (D) are called tracts of F over a; any set containing such a tract is called a neighborhood of this singularity.So we say that a sequence (z n ) in C converges to the singularity U if for every neighborhood U (D), all but finitely many members of this sequence belong to U (D).
If there exists D such that F (z) = a for all z ∈ U (D) (resp.such that F : U (D) → D \ {a} is a universal covering map), then the singularity, and the tract U (D), are called direct (resp.logarithmic).We note that there can be more than one transcendental singularity over the same point.The number of transcendental (or direct or logarithmic) singularities over a point a is just the number of different choices D → U (D).For example, the inverse of F (z) = exp exp z has infinitely many logarithmic singularities over both 0 and ∞, one logarithmic singularity over 1, and no other singularities.
We also note that F −1 has a transcendental singularity over a if and only if a is an asymptotic value of F .This means that there exists a curve γ tending to ∞ such that F (z) → a as z → ∞, z ∈ γ.Each neighborhood U (D) then contains a "tail" of this curve γ.
Langley's paper in fact contains the following generalization of Theorem A.
Theorem B. Let E be a real Bank-Laine function of finite order with only real zeros and let A and F be as in (1.4) and (1.5).
If A is non-constant, then the inverse F −1 has infinitely many logarithmic singularities over 0 and ∞, but the number m of singularities over points in C * := C \ {0} is finite.Moreover, we have the following: We will see that we actually have equality in these estimates of λ(E).
To state Theorem 1.3, we also recall that an entire function f of order ρ has completely regular growth in the sense of Levin and Pfluger if there exists a 2π-periodic function h f : R → R, not vanishing identically, such that as r → ∞, for re iθ outside a union of disks {z : |z − a j | < r j } such that (1.6) Our main result is the following theorem.
Theorem 1.3.Let F : C → C be a symmetric local homeomorphism with all zeros and poles real.Suppose that the number m of singularities of F −1 over points in C * is finite, but that F −1 has infinitely many singularities over 0 or ∞.
Then there exists a symmetric homeomorphism φ : For |θ| ≤ π, the indicator of E is given in case (i) by h E (θ) = c cos ρθ with c ∈ R \ {0} while in case (ii) we have h E (θ) = c sin(ρ|θ|) with c > 0 if the zeros are positive and h E (θ) = c sin(ρ|θ − π|) with c > 0 if the zeros are negative.In case (iii), h E is given by the (now coinciding) formulae of case (ii).In all cases we have with some c > 0. (v) All values of m indicated in (i) − (iii) can actually occur.Remark 1.2.All assumptions of Theorem 1.3 are of purely topological nature.So Theorem 1.3 contains a parabolic type criterion for a class of surfaces spread over the sphere.It can be compared with the theorem of Nevanlinna [45] describing the conformal type and asymptotic behavior of a locally univalent function F whose inverse has only finitely many singularities.In particular, Nevanlinna showed that a meromorphic function F with this property has finite order.
Suppose that F −1 has only finitely many singularities over 0 and ∞, but that F otherwise satisfies the hypotheses of Theorem 1.3.Then F −1 has only finitely many singularities, so belongs to the class considered by Nevanlinna.A result of Hellerstein, Shen and Williamson [26,Theorem 2] says that if all zeros and poles of a real meromorphic function F of this class are real, then F is a linear-fractional transformation or of the form F (z) = A tan(az + b) + B with real constants a, b, A, B.
The reality of zeros and poles of F is an essential assumption here and in Theorem 1.3:The results of [8] show in particular that there exist locally univalent meromorphic functions F whose inverses have only one singularity over C * , where the order of E = F/F can take any preassigned value in (1, ∞]. Remark 1.3.Theorem 1.3 is stronger than Theorem 1.2 for several reasons.First, Theorem 1.3 does not require the a priori assumption that E has finite order, but only the assumption that F −1 has finitely many singularities over points in C * .Second, we obtain a more precise description of the asymptotics of the functions E and A, namely that these functions are of completely regular growth.In particular, the functions are of normal type of the given order, a conclusion that does not follow from our analytic proof of Theorem 1.2; see Remark 3.3 below.Our proof of Theorem 1.3 will in fact give additional insights in the structure of these functions.Finally, the geometric approach also allows us to construct examples showing that all indicated values of m may actually occur.Note that Langley's Theorem A gives such examples for m = 2 in case (ii) and for m = 4 in case (iii).Some of the underlying ideas of the construction of our examples for general m are similar to his, but the details are quite different.It is plausible that Langley's methods could also be modified to yield examples for arbitrary m.
Remark 1.4.The case of arbitrary (not necessarily real) BL functions E of finite order with all zeros real can be reduced to the case of real BL functions by the following remark of Langley [36, p. 228]: Write E = Πe P +iQ , where Π is a canonical product with real zeros, and P and Q are real polynomials.Then condition (1.3) implies that at every zero z of E we have Π (z)e P (z)+iQ(z) = ±1.Since Π (z) and P (z) are real, we conclude that Q(z) ∈ πZ for every zero z of E. So Πe P is a real BL function with all zeros real.Furthermore, if E is real, then A is also real by (1.4), and F in (1.5) can be chosen real.Thus it suffices to consider only real functions F , E and A.
Remark 1.5.The Speiser class S is defined as the set of all meromorphic functions F : C → C for which there exists a finite subset A of C such that F : C \ F −1 (A) → C \ A is an (unramified) covering.It plays an important role in value distribution theory [22] and holomorphic dynamics [4,20,52].
Langley's Theorem B says in particular that if E is a BL function of finite order, then the associated locally univalent function F is in S.
Theorem 1.3 gives a description of real locally univalent functions F of class S with only real zeros and poles, for which the inverses have finitely many logarithmic singularities over values in C * , and infinitely many logarithmic singularities over each 0 and ∞.Since class S is much studied, this is of independent interest.Remark 1.6.Our proof of Theorem 1.3 uses topological arguments, quasiconformal surgery and the Teichmüller-Wittich-Belinskii theorem.These methods are frequently used to construct examples of meromorphic functions.In this paper, we also use this technique to prove a positive result.
We have discussed the equation (1.1) under the hypothesis that there are two solutions with only real zeros.Our final result addresses the case that there are three solutions with this property.Theorem 1.4.Let A be an entire function and suppose that (1.1) has three pairwise linearly independent solutions which have only real zeros.Then A is constant.Remark 1.7.Our starting point was [26,Theorem 3] which says that if A is a polynomial and if (1.1) has a basis of solutions with only real zeros, then A is constant.An extension of this result to linear differential equations of higher order has been given by Brüggemann [12,Theorem 5] and Steinmetz [53,Corollary 2].It would be of interest to which extent our results generalize to equations of higher order.This paper is organized as follows.Theorem 1.1 is proved in section 2. In section 3, we give a purely analytic proof of Theorem 1.2.The proof of Theorem 1.3 given in the subsequent sections is independent of this.In section 4 we collect the necessary prerequisites on the pasting-and-gluing techniques and line complexes, to make this paper self-contained.A reader familiar with this technique may pass to section 5, where we construct examples (Part (v) of Theorem 1.3) and outline the proof of all other parts.These parts are then proved in sections 6-8.Theorem 1.4 is proved in section 9.

Proof of Theorem 1.1
We use the standard notation of Nevanlinna theory as given in [22] or [46].The following result is due to Miles [43].
Lemma 2.1.Let f be an entire function of infinite order and suppose that the zeros of f lie on finitely many rays emanating from the origin.Then there exists a set L ⊂ [1, ∞) of logarithmic density zero such that Proof of Theorem 1.1.It follows from (1.4) and the definition of the proximity function Suppose that E has infinite order.Lemma 2.1 and the first fundamental theorem yield that there exists a set L of logarithmic density zero such that m r, On the other hand, the lemma on the logarithmic derivative [22, Chapter 3, § 1], applied to both E and E , implies that there exists a set M of finite logarithmic measure such that Combining the last three equations we conclude that Since A has finite order by hypothesis, this contradicts the assumption that E has infinite order.
3. Analytic proof of Theorem 1.2 Throughout this section, we consider a Bank-Laine function E and the functions A and F given by (1.4) and (1.5).To prove Theorem 1.2, we wish to establish upper and lower bounds on the behavior of E on the real axis.We begin by proving an upper bound.
and relate the singularities of G −1 over ∞ to the singularities of F −1 over non-zero finite values.Langley [37, Proposition 2.1, (C)] proved that under the hypotheses of Proposition 3.1 every neighborhood of a transcendental singularity of F −1 over a nonzero finite value contains a neighborhood of a singularity of G over ∞.We strengthen this result as follows.
Proposition 3.2.Let E be a Bank-Laine function of finite order, F a locally univalent function satisfying Then there is a bijection between the singularities of G −1 over ∞ and the singularities of F −1 over values in C * , with the following property: Any sequence of points converging to a singularity of G −1 over ∞ also converges to the corresponding singularity of F −1 .
In order to prove Proposition 3.2 we will use the following lemma.Lemma 3.3.Let F be a meromorphic function and set E := F/F .Then every neighborhood of a direct transcendental singularity of E −1 over ∞ contains an asymptotic path for some asymptotic value a ∈ C * of F .
To prove the lemma, we use the following result of Huber [30]; see also [41].Then there exists a path γ tending to ∞ such that γ e −λu(z) | dz| < ∞.
Proof of Lemma 3.3.Let W be a neighborhood of a direct singularity of E over ∞; we may choose W as a component of {z : E(z)| > K} for some K > 0. Assuming that K is large, W is a direct tract.This implies that the function This means that the image of γ under a branch of log F has finite Euclidean length, and hence this branch tends to some value β ∈ C as z tends to ∞ along γ.Setting α := e β ∈ C * we thus have When F is locally univalent, then E is entire and hence every transcendental singularity of E over ∞ is direct.In particular, every neighborhood of a transcendental singularity of G over ∞ is also a neighborhood of a direct singularity of E, and thus contains an asymptotic path of F −1 .Under the hypotheses of Proposition 3.2, the corresponding singularity of F −1 is logarithmic by [35,Corollary 1.1].To prove Proposition 3.2, we use an estimate on the derivative of a function having a logarithmic singularity, which is a consequence of Koebe's theorem.Such estimates are useful in other contexts, notably in the study of the class B in complex dynamics, and therefore we state the result in this generality for future reference.(A similar estimate is also used by Langley; see the second displayed formula in the proof of [37, Proposition 2.1].)Lemma 3.5.Let H = {z : Re z > 0} be the right half-plane and φ : Proof.Pre-and post-composing by suitable affine maps, we may assume that z 0 = 1 and φ (z 0 ) = 1.Put M (z) := (z − 1)/(z + 1).Then M maps H conformally to the unit disk and we have for Re z ≥ 1.
Proof of Proposition 3.2.Let E, F and G be as in the statement of the Proposition.Let α ∈ C * be an asymptotic value of F .As already mentioned, every singularity ξ of F over α is logarithmic.Claim.If U is a sufficiently small neighborhood of ξ, then |G(z)| is bounded on ∂U .Moreover, for large enough R > 0, the set {z : |G(z)| > R} ∩ U is unbounded and connected.
To prove the claim, observe first that 1/G(z) is the derivative of ζ → log F (exp(ζ)), where exp ζ = z.To study this in more detail, choose β with exp β = α and put D := D(β, ε) for some small ε > 0.Here and in the following D(β, ε) denotes the open disk of radius ε around β. Let Ω be the connected component of F −1 (exp(D)) that is a neighborhood of ξ.If ε is sufficiently small, then F : Ω → exp(D) \ {α} is a universal covering and 0 / ∈ Ω.Let T be a connected component of exp −1 (Ω).Let λ be the branch of (log F ) • exp on T that takes values in D \ {β}; then λ is also a universal covering.If ζ ∈ T and z = exp(ζ), then If φ is a conformal map from H onto T , then λ•φ : H → D\{β} is a universal covering.Another universal covering from H onto D \ {β} is given by w → ε exp(−w) + β.We may normalize φ so that these two maps are equal.Thus for w ∈ H. Since T is disjoint from its 2πiZ-translates, we have Re w for all w ∈ H by Koebe's theorem.(See [20, Lemma 1], and compare [47].)Thus, by (3.2), |g(w)| is bounded when Re w is bounded away from 0 and ∞.Note also that if w ∈ H with Re w = R > 0, then z = exp φ(w) satisfies The first part of the Claim follows for the component U of F −1 (exp(D(β, εe −R ))) contained in Ω, and in fact for every sufficiently small neighborhood U of ξ.Then Re w > 1 for all w ∈ V by (3.2) and (3.3), and hence |g(w)| = R for all w ∈ ∂V .Since g is unbounded on V there is some So V contains a sector based at w 0 as above, and in particular all sufficiently large points at argument between −π/5 and π/5.Hence the component V is unique, and the Claim is proved.
To complete the proof, recall that, since E and hence G are of finite order, the number n of singularities of G −1 over ∞ is finite.Let K 0 > 0 be so large that {z : |G(z)| > K 0 } has exactly n unbounded components, one component V (S) for each singularity S of G −1 .By Lemma 3.3, there is a singularity S of F −1 over some value α ∈ C * such that every neighborhood of S intersects V (S).By the Claim, every neighborhood of S is also a neighborhood of S, and hence any sequence of points converging to S also converges to S .That the map S → S is a bijection also follows from the Claim.
Proof of Proposition 3.1.Suppose that E has infinitely many positive zeros, and let Ω be a logarithmic tract of F over a point in C * .We may assume that Ω contains no zeros and poles of F and hence no zeros of E. Since a logarithmic tract is simply connected and since a logarithmic tract intersecting the real axis is symmetric, the intersection of Ω with the real axis is connected, and hence bounded from above.So if (x n ) is a sequence tending to +∞, then (x n ) does not converge to a transcendental singularity of F −1 .Hence, by Proposition 3.2, a sequence (x n ) tending to +∞ cannot converge to a transcendental singularity of G −1 .Thus |G(x n )| is bounded for such a sequence.In other words, |G| is bounded on the positive real axis, as claimed.
The case that E has infinitely many negative zeros reduces to the case of positive zeros by considering E(−z) instead of E(z).
Having established an upper bound for G and hence E on the real axis, we now prove a lower bound, outside certain neighborhoods of the zeros.We use the following lemma due to Laguerre and Borel; see [42].Lemma 3.6.Let f be a real entire function of finite genus p with m non-real zeros.Then, in addition to one zero of the derivative f of f between each pair of adjacent real zeros of f , the derivative f has at most p + m real and non-real zeros.
Here zeros are counted with multiplicities.The result implies that a real entire function of finite order with only finitely many non-real zeros has only finitely many local minima where f (x) ≥ 0 and only finitely many local maxima where f (x) ≤ 0.Moreover, the same is true for all its derivatives.Thus we can apply the following fact to the restriction of E to the interval between any two successive (and sufficiently large) zeros.(e) f is negative at every local minimum and positive at every local maximum.Then We first claim that for 0 ≤ x ≤ 1/3, and hence Without loss of generality, we may assume that c ≥ 1/2.Otherwise, replace f by It follows that f (τ ) takes a local maximum between 1/3 and 2/3.By assumption (e), this maximum value is positive.So the interval [η, η * ] contains at least two zeros of f , bounding an interval on which f is positive.Since f (c) ≤ 0, these zeros either both belong to [0, c] or both to [c, 1], contradicting (d).Thus (3.6) is proved.
To complete the proof of the lemma, we show that The minimum on the left-hand side is assumed either at x = ε or x = 1 − ε; we may assume the former.In particular, c ≥ ε.By the mean value theorem, there exists ξ ∈ (0, ε) such that There also exists ξ * ∈ (ε, c) such that We may assume that f (ε) ≤ ε since otherwise there is nothing to prove.Then f (ξ) ≤ 1.
Remark 3.2.The lower bound for |f (x)| in (3.5) can certainly be improved, but it suffices for our purposes.
Given an entire function f , a sequence (r k ) is called a sequence of Pólya peaks of order σ ∈ [0, ∞) for log M (r, f ), where M (r, f ) = max |z|=r |f (z)| is the maximum modulus, if for every ε > 0 we have Drasin and Shea [14] proved that Pólya peaks of order σ exist for all finite σ ∈ [ρ * , ρ * ] and that we always have where µ(f ) denotes the lower order of f .
Proof of Theorem 1.2.We will first consider the case that E has infinitely many positive and infinitely many negative zeros; that is, we will prove conclusion (b) of the theorem.The minor modifications to handle conclusion (a) will be discussed at the end of the proof.
We will show that if (r k ) is a sequence of Pólya peaks of some order σ for log M (r, E), then σ = N for some odd integer N .In view of (3.10) this yields that µ(E) = ρ(E) = N .
We consider the subharmonic functions u k given by Given ε > 0 we then have ε and large k.This implies (cf.[28, Theorems 4.1.8and 4.1.9]or [29, Theorems 3.2.12 and 3.2.13])that, passing to a subsequence if necessary, the limit exists and is either −∞ or a subharmonic function in C.Here the convergence is in the Schwartz space D .This implies that we also have convergence in L 1 loc .There are a number of papers where entire and meromorphic functions are studied in terms of a subharmonic u obtained as in (3.11); see, e.g., [17] for further details.
The function u is harmonic in C \ R and satisfies We shall show that we also have u(x) ≥ 0 and thus u(x) = 0 for x ∈ R. In order to do so, suppose that u(x) < 0 for some x ∈ R. Then there exist δ > 0, t > 0 and Suppose that ξ > 0, and let (x n ) be the sequence of positive zeros of E. (The case ξ < 0 is analogous, replacing positive by negative zeros).We choose n such that for sufficiently large k.This contradicts (3.14); thus Since u is harmonic in the upper half-plane, the reflection principle for harmonic functions yields that This implies that w is a polynomial and in fact that σ ∈ N and w(z) = cz σ for some By (3.13) we have |c| = 1.Since u(x) = v(x) = Im w(x) = 0 for x ∈ R we have c ∈ R and thus c = ±1.Hence we have v(re iθ ) = cr σ sin(σθ) and consequently u(re iθ ) = cr σ sin(σ|θ|) for |θ| ≤ π.Since u is subharmonic, u cannot have local maxima.In particular, there are no local maxima on the positive ray.This yields that c = 1.Then, since there are no local maxima on the negative ray, we conclude that σ is odd, say σ = 2n − 1 with n ∈ N. Thus Moreover, it follows from (1.4) and the lemma on the logarithmic derivative [22, Chapter 3, We conclude that we also have ρ(A) = 2n − 1. Theorem A yields that n ≥ 2. This completes the proof of (b).

It follows that
As before we can now conclude that λ Remark 3.3.One can derive from this proof further regularity of the asymptotic behavior of A and E. Indeed, since we established that the possible values of the order σ of the Pólya peaks form a discrete sequence, we conclude that where is a slowly varying function in the sense of Karamata; that is, (cr)/ (r) → 1 as r → ∞ for every c > 1.Moreover, outside an exceptional set satisfying (1.6).Similar statements apply to the function A.
However, as we mentioned in Remark 1.3, the method of this section does not allow to show that A and E are of normal type.
Remark 3.4.In the proof of Theorem 1.2 we used Theorem A to conclude that n ≥ 2. This can also be seen directly.In fact, suppose we have n = 1 in case (b).Then (3.18) takes the form log |E(re iθ )| ∼ r (r) sin |θ|.
To estimate the logarithmic derivative of E we use the Schwarz integral formula.It says that if g is holomorphic in a domain containing the closed disk D(a, t), then for z ∈ D(a, t).It follows that Since the zeros of E are real, we may apply this to a branch g of log E in the upper or lower half-plane.Noting that E /E = g and E /E = g + (g ) 2 we find that given ε > 0 there exists 4.1.Gluing of elements.As a general reference for the concepts considered in this section we mention [18].We consider connected bordered oriented surfaces D, not necessarily compact.The border ∂D is equipped with the standard orientation (such that the interior stays on the left).
An element (D, f ) is a pair, where D is a bordered surface and f : D → C is a continuous function that is topologically holomorphic on the interior of D and locally injective near every point of the border ∂D.Two pairs (D 1 , f 1 ) and (D 2 , f 2 ) are called equivalent if there is an orientation-preserving homeomorphism φ : There is a unique conformal structure on D that makes f holomorphic.Equivalence classes are called Riemann surfaces spread over the sphere.By the Uniformization Theorem, each pair (D, f ) with open simply connected D 0 is equivalent to a pair (D 0 , f 0 ) where D is the whole plane or an open disk and f 0 is meromorphic in D 0 .
Consider two elements (D 1 , f 1 ) and (D 2 , f 2 ) and suppose that there are two closed arcs I j ⊂ ∂D j which are mapped by f j onto the same arc homeomorphically, but with opposite orientations.Recall that, by definition of an element, the f j have no critical points on I j .Then there is a homeomorphism φ : By gluing D 1 and D 2 along this homeomorphism we obtain a new element (D, f ), where f = f j on D j .(For the formal definition of gluing of topological spaces see, for example, [11, § 2.5].)We can apply this procedure when (D 1 , f 1 ) = (D 2 , f 2 ), and glue an element to itself.
The following result says that this gluing operation is compatible with the equivalence relation on elements.
), and we can glue (D 1 , f 1 ) to (D 3 , f 3 ) along some arcs, then (D 2 , f 2 ) can be glued to (D 4 , f 4 ) along the corresponding arcs, and results of the gluings are equivalent.
Proof.Let φ 1 : D 1 → D 2 and φ 3 : D 3 → D 4 be homeomorphisms as in the definition of equivalence and let ψ 1 : I 1 → I 3 , where I 1 ⊂ ∂D 1 and I 3 ⊂ ∂D 3 , be the gluing homeomorphism.Thus ), and verify that So (D 2 , f 2 ) and (D 4 , f 4 ) can be glued along the arcs φ 1 (I 1 ) and φ 3 (I 3 ).Now the homeomorphism φ from the gluing of D 1 and D 2 to the gluing of D 3 and D 4 is given by the formula One checks using (4.1) that ψ 1 and ψ 2 match on the arc along which D 1 and D 2 were glued; that is, we have If one glues two simply connected surfaces along a connected arc one obtains a simply connected surface.If one glues one simply connected surface to itself along two disjoint arcs, one obtains a doubly connected surface of genus zero which is homeomorphic to a ring in the plane.Every doubly connected open Riemann surface of genus zero is conformally equivalent to a round ring {z : r < |z| < R}, where 0 ≤ r < R ≤ +∞.

Cell decompositions.
A cell decomposition of a bordered surface D is a representation of D as a locally finite union of disjoint cells of dimensions 0 (vertices), 1 (edges) and 2 (faces), so that the boundary of each cell is a union of cells of smaller dimension.Here a vertex is a point.An edge is homeomorphic to an open interval and its closure is homeomorphic to a closed interval.A face is homeomorphic to an open disk and its closure is homeomorphic to a closed disk or closed half-plane.
The edges and vertices of a cell decomposition can be of two types: those which belong to ∂D we call boundary edges (vertices), and those which belong to the interior of D we call inner edges (vertices).
Two cell decompositions are combinatorially equivalent if there is a bijection p of the set of cells of one of them onto the set of cells of the other, respecting the cell dimensions, and such that p(∂c) = ∂p(c) for every cell c.This is equivalent to the existence of a homeomorphism of the ambient surfaces which maps each cell homeomorphically.We call such pairs of cell decompositions simply equivalent.

Labeled cell decompositions.
Let us consider a cell decomposition C of C, with two vertices which we denote × and •.Suppose that there are q ∈ N edges, with q ≥ 2, each edge connecting × to •, and q faces.All faces are digons.Suppose that a finite set A containing one point in the interior of each face is given.Let (D, f ) be an element and let L be a cell decomposition of D such that f maps every cell of L into a cell of C of the same dimension, with the following properties: (a) The restriction of f onto the closure of each edge of L is a homeomorphism onto the closure of an edge of C, (b) For each face c of L, the restriction of where {a} = A ∩ f (c).It follows from (b) that each interior edge of L belongs to the boundaries of two distinct faces.A boundary edge evidently belongs to the boundary of one face.
We label each vertex of L by × or •, according to its image, and we label each face of L by the element of A which is contained in its image cell.This set of labels defines the image under f of each cell of L. Indeed, a boundary edge can be labeled by the label of the unique face to which it belongs, and an inner edge can be labeled by the pair of labels of two faces to whose boundaries it belongs.Since every two faces of C have at most one common boundary edge, the image of each edge of L is determined by the labels of faces and vertices.Proposition 4.2.Let C and A be as above, and (D 1 , f 1 ), (D 2 , f 2 ) be two elements with labeled cell decompositions L 1 , L 2 of D 1 , D 2 satisfying conditions (a) and (b).If L 1 is combinatorially equivalent to L 2 , where the equivalence respects the labels, then Proof.We have to define a homeomorphism φ : be the bijection between cells of L 1 and L 2 .Since this bijection preserves labels, for each cell c of L 1 , we have f 1 (c) = f 2 (c ).This defines a unique homeomorphism φ of the 1-skeleton of L 1 onto the 1-skeleton of L 2 such that f 1 = f 2 • φ on the 1-skeleton.It remains to extend φ into faces.Let c and c be closures of some corresponding faces of L 1 and L 2 .Let a ∈ A be the label of c.Let z 0 ∈ ∂c, and w 0 = φ(z 0 ) ∈ ∂c .Let z ∈ c, and choose an open arc γ in c from z 0 to z such that a ∈ f 1 (γ).There is a unique lift γ 2 of this curve by f 2 starting at w 0 .The endpoint w of γ 2 determines φ(z).It is evident that thus defined φ is the required homeomorphism.

Representation of Speiser class functions by line complexes and trees.
The definition of the Speiser class S given in Remark 1.5 extends to topologically holomorphic maps.So we say that a topologically holomorphic map f : For a local homeomorphism we can choose A to be the set of asymptotic values.When f is a local homeomorphism, all faces of L f are either digons or ∞-gons.The restriction of f on each digon is a homeomorphism, while the restriction of f on an ∞-gon is a universal covering of a face of C minus the element of A which it contains.
For a local homeomorphism f , one can describe the cell decomposition L f by a simpler object.If for some pair of vertices there is more than one edge connecting them, replace all these edges by a single edge.So digons disappear.We preserve the labels of the remaining faces and the labels of vertices.The resulting cell decomposition T = T f has the following properties: (i) Its 1-skeleton T is a bipartite graph embedded in C.
(ii) The cyclic order of face labels around each ×-vertex (•-vertex) is consistent with (a restriction of) the cyclic order of face labels around the ×-vertex (•-vertex) of C. Given any connected bipartite graph embedded in C with labeled complementary components satisfying (ii), one can recover the whole line complex L in a unique way (up to equivalence).
In the case that f is a local homeomorphism, T f is a tree.So equivalence classes of local homeomorphisms f of class S are encoded by trees embedded in C with labeled vertices and complementary components satisfying (ii).4.5.Symmetric local homeomorphisms.We recall that a local homeomorphism f : In this case, the set of asymptotic values is symmetric, that is, we have A = A. A cell decomposition is called symmetric if the complex conjugation z → z maps each cell onto a cell of the same dimension with complex conjugate label.If f and the cell decomposition C are symmetric, then the line complex L f and the tree T f are symmetric.Conversely, to each symmetric C and L (or T ) corresponds a symmetric local homeomorphism f .
The complex conjugation acts on the faces of a symmetric cell decomposition.So each face of a symmetric cell decomposition is either symmetric or disjoint from the real line.
Unfortunately, for a symmetric set A, a symmetric cell decomposition C may not exist.(It exists if and only if A contains at most 2 real points.)There are several methods of dealing with this difficulty in the study of symmetric functions of class S; see, for example, [21,19].In this paper we will use a somewhat different approach, by using symmetric sub-decompositions of a generally non-symmetric C. 4.6.Quasiconformal mappings.While the first part of the proof of Theorem 1.3 given in section 7 will only deal with topologically holomorphic mappings, the second part given in section 8 will also use quasiregular and quasiconformal mappings.For the definition and general properties of quasiconformal mappings we refer to [3,39].We note that quasiregular mappings are called quasiconformal functions in [39].
For a region D and a quasiregular map f : D → C we use the notation The following result is a consequence of the existence theorem for a quasiconformal mappings with prescribed dilatation [39, § V.1].
Lemma 4.4.Let U and V be neighborhoods of ∞ and let φ : U → V be a quasiconformal map with φ(∞) = ∞.Suppose that The logarithmic area of a measurable subset A of C is defined by then (4.2) and hence (4.3) hold.
The next lemma is easily proved by direct computation.
Lemma 4.5.Let A ⊂ C be measurable and α > 0. Suppose that some branch of z → z α is injective on A and let A α be the image of A under this branch.Then In particular, logarea(A α ) is finite if and only if logarea(A) is finite.

Construction of examples, and outline of the proof of Theorem 1.3
In this section, we will prove statement (v) of Theorem 1.3.Note that we only need to construct examples of local homeomorphisms satisfying the assumptions of the theorem with any m as described in (i) − (iii).In order to conclude that all values of m do actually occur for meromorphic functions, we will need the first part of the theorem which says that there is a homeomorphism φ such that F 0 = F • φ is a meromorphic function.This part will be proved later.The inverses of the functions g m and h m , as well as those of all other functions we construct in this section, clearly have infinitely many singularities over 0 and ∞.

5.2.
Infinite one-sided sequence of zeros and poles.The required function with m ≥ 2 is constructed by gluing two elements.Our first element is (D 1 , f 1 ) where D 1 = {z : Re z > 0} is the right half-plane and f 1 (z) = tan z.
The second element is very closely related to the function h k introduced in (5.1) with k = 2(m − 1).For m ≥ 2 we put t dt and define (5.2) Our second element is (D 2 , f 2 ) where D 2 = {z : Re z < 0} is the left half-plane.
Both f 1 and f 2 map the imaginary axis homeomorphically onto the interval (i, −i), with the same orientation.Viewed as the boundary of D 1 and D 2 , the imaginary axis has opposite orientations.Thus (D 1 , f 1 ) and (D 2 , f 2 ) can be glued.The resulting function f is locally univalent and has m singularities over C * .Moreover, the gluing can be done symmetrically so that the resulting function f is also symmetric, with all zeros and poles on the positive axis.The resulting function has f has n poles and n + 1 zeros.Of course, 1/f has n + 1 poles and n zeros.
In order to construct an example with the same number n of zeros and poles, one takes D 1 = {z : 0 < Re z < (n − 1/2)π} and glues 1/f 2 to it along the line {z : Re z = (n − 1/2)π}.
The number of singularities over C * in such examples is even.To obtain an odd number greater than 1, we can use functions f 1 and f 3 with different values of m.It seems that there are no examples with m = 1 and a finite number larger than 1 of zeros and poles.5.4.Infinite two-sided sequence of zeros and poles.These examples use the line complex and the associated tree, as described in section 4. We will have four asymptotic values in C * , say {i, −i, 2i, −2i}.
We consider a symmetric cell decomposition C of C with two vertices × and • on the real line, such that each face contains exactly one asymptotic value.Then we consider the tree T embedded in C as shown in Figure 1 for m = 4.This tree satisfies all conditions stated in § 4.4, for the cell decomposition shown in Figure 2, so it defines a local homeomorphism F .Observe that every non-real vertex is adjacent to logarithmic tracts both over 0 and over ∞, which implies that all of the zeros and poles of F are real; hence F satisfies the hypotheses of Theorem 1.3 (iii), for m = 4. 2i i We may insert additional logarithmic tracts by splitting the tree at any vertex that is adjacent only to a face labeled 0 and a face labeled ∞ (and symmetrically, at its complex conjugate), inserting additional branches to ensure that every non-real vertex is again adjacent to a face labeled 0 and a face labeled ∞.See Figure 3. Repeating this procedure, we obtain local homeomorphisms with the desired properties for every even m ≥ 4, as claimed.5.5.Outline of the proof of Theorem 1.3.We will show that all functions F satisfying the conditions of Theorem 1.3 are similar to these examples.To do this, we draw m disjoint asymptotic curves γ j corresponding to all distinct singularities of F −1 over C * .These curves will split the plane into m disjoint sector-like regions G 0 , . . ., G m−1 plus a compact set.Assuming that F has infinitely many positive zeros and poles, we enumerate them so that the positive zeros and poles are contained in G 0 and that the whole partition of the plane into G 0 , . . ., G m−1 is symmetric.So when there are infinitely many negative zeros and poles, they will lie in G m/2 .
Then we will show that the restrictions of our function F onto G j are of two special types described in the next section.One type which we call T a is symmetric, has infinitely many zeros and poles, but 0 and ∞ are not asymptotic values for this element.This element is similar to the element (D 1 , f 1 ) considered in § 5.  has no zeros and poles, but infinitely many singularities over 0 and ∞.It is similar to the element (D 2 , f 2 ) of § 5.2, with m = 2.
In the next section we construct explicit quasiregular representatives of these classes of elements, and show that their quasiconformal dilatation is supported on a small set (of finite logarithmic area).This will allow us to paste them together to reconstruct our function F .All asymptotic properties will follow from the Teichmüller-Wittich-Belinskii theorem (Lemma 4.4).

Elements of special type
As explained in § 5.5, we will divide the plane into certain regions G j such that the restriction of F to G j is equivalent to one of two special types of elements.These elements (D, f ) have the following properties: The domain D is bounded by a curve γ : R → C such that |γ(t)| → ∞ as t → ±∞.Here γ is oriented in the positive direction so that D is on the left of γ.The function f is such that f (γ(t)) tends radially to certain asymptotic values as t → ±∞.This means that there exist t ± ∈ R, a ± ∈ C * , θ ± ∈ (−π, π] and ε > 0 such that ( * ) f • γ maps [t + , ∞) homeomorphically onto (a + , a + + εe iθ + ] and (−∞, t − ] homeomorphically onto (a − , a − + εe iθ − ].We will only need the cases where θ ∈ {0, ±π/2, π} so that f approaches the asymptotic value parallel to the real or imaginary axis. We call the pairs d ± := (a ± , θ ± ) oriented asymptotic values.If there exist t ± ∈ R and ε > 0 such that ( * ) holds, then we say that the element (D, f ) has the oriented 2i i i ∞ 0 asymptotic values d ± at ±∞.We shall assume that if a + = a − , then θ + = θ − , since this suffices for our purposes.
One element will essentially be the restriction of the tangent function to the right half-plane H := {z : Re z > 0}.The boundary curve γ H is then given by γ H (t) = −it.Then ( * ) holds with a ± = ∓i and θ ± = ±π/2.Thus (H, tan) has the oriented asymptotic values (∓i, ±π/2) at ±∞.For technical reasons, we will, however, later introduce some modification of the element (H, tan).Essentially, we will show that if F has infinitely many zeros and poles in G j , then (G j , F ) is equivalent to this (modified) element.
But before doing so we will deal with the domains G j where F has no zeros and poles.We will show that these elements can be represented in a particular form, similar to the ones considered in § 5.1.We begin with the case where a ± > 0 and θ ± = 0.Here and in the following we put H + = {z : Im z > 0}.
6.2.Let a ± > 0 and put d ± = (a ± , 0).Let (D, F ) be of type B d + ,d − .Then there exist compact sets K and K 0 , a symmetric rational function R 0 with at most one pole and ξ, c 0 ∈ R such that with (6.1) To prove Proposition 6.2, we will use the following lemma.We omit its proof, but note that it can easily be deduced from a result of Morse and Heins [44,Theorem 20.4].Lemma 6.3.Let 0 < t < T and let f : {z : |z| > t} → C be a topologically holomorphic map.Then there exists a topologically holomorphic map The map F can be chosen to have at most one pole, which is located at 0. Moreover, if f is symmetric, then F can be chosen to be symmetric.
The following result is due to Lindelöf [22, Chapter 5, Lemma 1.2].Lemma 6.4.Let f : {z : Re z ≥ 0} → C be continuous and bounded.Suppose that f is holomorphic in {z : Re z > 0} and that there exist a ± ∈ C such that f (it) → a ± as t → ±∞.Then a := a + = a − and f (z) → a as |z| → ∞.
We will also use the following result.Lemma 6.5.Let f be a meromorphic function with only finitely many zeros and poles.Suppose that there exists N ∈ N such that for each R > 0 the set {z : |f (z)| > R} has at most N unbounded components.Then f has the form where Q is rational and where P is a polynomial of degree at most N .
Proof.It is clear that f has the form (6.2) with a rational function Q and an entire function P .We have to prove that P is a polynomial of degree at most N .We may choose K so large that every connected component of A containing one of the finitely many poles of f is bounded, and such that A has the maximal number of unbounded connected components subject to this condition.Then each unbounded connected component of A is a neighbourhood of a unique singularity of f −1 over ∞.Similarly, for large K each unbounded connected component of B is a neighbourhood of a unique singularity of f −1 over 0. By Lindelöf's lemma 6.4, "between" two such singularities of f −1 over 0 there is a singularity of f −1 over ∞, and vice versa.Therefore, between two unbounded components of A there is an unbounded component of B, and vice versa.In particular, A and B have the same finite number of unbounded connected components for large K.
By [55,41] each unbounded component of A contains a path γ such that u(z)/ log |z| → ∞ as z → ∞, z ∈ γ, while each unbounded component of B contains a path γ such that u(z)/ log |z| → −∞ as z → ∞, z ∈ γ.Thus, given K > 0, the "tail" of such a curve γ is contained in a component of respectively.(Note that the components of A and B are always unbounded.)By the same argument, the components of A and B contain curves whose tails are contained in unbounded components of A and B, respectively.Overall we see that there is a oneto-one correspondence between the unbounded components of A and the components of A .We claim that A has connected complement if K is sufficiently large.In order to show this we note that every complementary component of A must contain some connected component of B , and thus the number of complementary components is finite.In particular, there is R > 0 such that, for sufficiently large K , every complementary component of A intersects the disk D(0, R).If additionally K > M (R, P ), then this disk does not intersect A , and therefore the complement of A is connected.This implies that the complement of every component of A is connected.
By hypothesis, A has at most N unbounded components, and hence A has at most N components.Let now w ∈ C with Re w > K such that w is not a critical value of P and let z 1 , z 2 ∈ C be such that P (z 1 ) = P (z 2 ) = w.We will show that if z 1 = z 2 , then z 1 and z 2 are contained in different components of A .Thus w has at most N preimages.Since this holds for every w with Re w > K which is not a critical value of P , the conclusion follows.
Suppose that now that z 1 and z 2 are in the same component U of A .We connect z 1 and z 2 by a simple curve γ 0 in U which intersects γ 1 and γ 2 only at z 1 and z 2 .Since U has connected complement, the curves γ 0 , γ 1 and γ 2 bound a subdomain V of U .Its image P (V ) is an unbounded domain whose boundary is contained in the union of the ray {w + re it : r ≥ 0} with P (γ 0 ).This implies that P (V ) contains points with real part less than K .This is a contradiction since V ⊂ U and U is a component of A .
Proof of Proposition 6.2.Without loss of generality we may assume that F is holomorphic, D = H + and γ(t) = t.Since (D, F ) has the oriented asymptotic values (a ± , 0) at ±∞, there exists a compact interval I such that, by reflection, F extends to a map holomorphic in C \ I.By Lemma 6.3 there exist R > 0 and a symmetric, topologically holomorphic map Here φ 1 can be chosen to be symmetric so that F 0 is also symmetric and has no poles in C * .
The function F 0 has only finitely many zeros and no poles in C * .Thus there exist a symmetric rational function Q and a symmetric entire function g such that z) .
Moreover, F 0 has only finitely many critical points.Thus has only finitely many zeros.Hence there exist a symmetric rational function R 0 with no poles in C * and a symmetric entire function h such that z) .
Note that F 0 has the form (6.4) for some c 0 ∈ R. The conclusion thus follows if we can show that h in (6.3) is a quadratic polynomial, since once this is known, we can achieve by an affine change of variable that h(z) = −z 2 .This will change the lower limit in the integral in (6.4) from 1 to some ξ ∈ R.
For K > 0, consider a component U of Hence there exists α ∈ C such that if z is a point on Γ and if Γ z denotes the part of Γ which connects the starting point of Γ with z, then Hence there exists By hypothesis, the inverse of F and hence that of F 0 have no singularities over values in C * except for the two asymptotic values a ± , for which the positive and negative real axis are asymptotic paths.
This implies (cf.Proposition 3.2) that the set considered in (6.5) has at most two components which do not contain a zero of L. Lemma 6.5 implies that h is a polynomial of degree at most 2. In fact, since the positive and negative axis are asymptotic paths, the degree of h is exactly 2. Proposition 6.2 required that a ± > 0 and θ ± = 0. We will now lift this restriction.In our applications, it will be convenient to work with the slit plane Ω 0 := C \ [0, ∞) instead of the upper half-plane H + .Note that z → z 2 maps H + onto Ω 0 .The boundary of Ω 0 is parametrized by the curve γ Ω : R → ∂Ω which is given by γ Ω (t) = −t for t < 0 and γ Ω (t) = t for t ≥ 0. So we consider the bordered surface Ω whose interior is Ω 0 and the border ∂Ω is the curve γ Ω .We put ∆ = {z : |z| ≥ 1}.Proposition 6.6.Let (D, F ) be of type B d + ,d − .Then there exist compact sets K and K , a local homeomorphism B : Ω \ K → C and t 0 > 0 such that (Ω \ K , B) ∼ (D \ K, F ) and The map B is quasiregular with and there exists c ∈ C * and d ∈ R with 2d ∈ Z such that, as z → ∞, in any closed subsector of the first or fourth quadrant, respectively, while in any closed subsector of the left half-plane.
Proof.As already mentioned, we will reduce this result to Proposition 6.2.Since this proposition is phrased for functions defined in the half-plane H + , we will first prove a "half-plane version" of our conclusion.We will show that there exist compact sets K and K 1 , a local homeomorphism F 1 : H + \K 1 → C and x 0 > 0 such that (H + \K 1 , F 1 ) ∼ (D \ K, F ) and (6.10) Here F 1 is quasiregular with Moreover, there exists c ∈ C * and d ∈ Z such that if δ > 0, then, as z → ∞, (6.12) we then see that B has the properties stated.
In order to construct the function F 1 we note that if a + = a − , then there exist ε > 0 and a quasiconformal map ψ : C → C such that If a + = a − := a and thus θ + = θ − =: θ, then we choose ψ such that In the first case (D, ψ −1 • F ) has the oriented asymptotic values (2 ±1 , 0) at ±∞, in the second case (D, ψ −1 • F ) has the oriented asymptotic value (1, 0) at ±∞.
In both cases, Proposition 6.2 is applicable to (D, ψ −1 • F ) and we conclude that there exists a function F 0 as given in (6.1) and compact sets K and K 0 such that In order to prove (6.10) and (6.11), we restrict to the case a + = a − .The other case is analogous.Then F 0 has the oriented asymptotic value (2, 0) at +∞ and thus exp For large x, say x ≥ x 0 , we then have Similarly, for negative x of large modulus, and we may choose x 0 such that this holds for x ≤ −x 0 .Note that since F and ψ • F 0 have the oriented asymptotic values (a ± , θ ± ), this implies that R 0 (x) < 0 if |x| is large.We may assume that this holds for |x| ≥ x 0 .We put We have u(x) > 0 and can thus define h : and hence (6.16) Moreover, a computation shows that (6.17) h (x) → 1 as x → ±∞.
Increasing the value of x 0 if necessary, we can extend h to a diffeomorphism of R. We now define τ : For 0 < y < 1 we have .
Using (6.16) and (6.17) we see that τ is quasiconformal.We now put for x ∈ R we deduce from (6.14) and (6.15) that for ±x ≥ x 0 .Thus we have (6.10).
To prove (6.11) let Recall here that F 0 has logarithmic singularities over 2 ±1 , as well as over 0 and ∞.Thus M is disjoint of the set of singular values.To prove (6.11) it suffices to prove that In the terminology of [16, Definition 1.5] we thus have to show that F 0 has the area property. Put We will show that (6.20) logarea(F −1 0 (M ) ∩ T j ) < ∞ for 1 ≤ j ≤ 3.This easily yields (6.19).
Since F 0 (x) → 2 as x → ∞ we find that g(x) → log 2 as x → ∞.Similarly as before we find that there exists r 0 such that Here the path of integration connects z with the positive axis, and then runs to ∞ along it.To be definite, let the path of integration be the segments [z, 2|z|] and [2|z|, ∞).
To show that (6.20) also holds for j = 2, we note that z → p(z) := i √ z maps the right half-plane H conformally onto {z : π/4 < arg z < 3π/4}.Put g 0 = g • p and G 0 = F 0 • p = exp g 0 .In view of Lemma 4.5 it thus suffices to show that with Integration by parts yields that (6.24) Thus there exists α ∈ C * and β ∈ R such that For z ∈ T 2 of sufficiently large modulus we may thus write g 0 (z) = exp ϕ 0 (z) with a map ϕ 0 satisfying Hence for C ⊂ T 2 we have logarea ϕ 0 (C) < ∞ if and only if logarea C < ∞.To prove (6.23) it thus suffices to show that has finite logarithmic area.In order to do so we put, for k ∈ Z, For k ≥ 1 and z = x + iy ∈ S k we have The same argument yields that if This completes the proof of (6.23) and hence (6.22), finishing the proof of (6.10).Finally, it follows from the definition of F 1 and (6.21) that This yields the asymptotics for F 1 (z) − a + given in (6.12).Those for F 1 (z) − a − are obtained analogously.Finally, (6.13) follows from (6.24) in the same fashion.This completes the proof of Proposition 6.6.

6.2.
Elements with infinitely many zeros and poles.We want to glue the element (Ω \ K , B) from Proposition 6.6 to (a modification of) the tangent map.Instead of the tangent we will, for a ∈ C \ R, consider the restriction of (6.To make this gluing explicit we modify the map v a .Note that as well as Our aim is to construct a quasiconformal map φ such that if (6.26) holds, then there exists t 1 ≥ t 0 such that respectively.
Definition 6.7.An element (D, F ) is of type T a if there exist compact sets K and K such that (D \ K, F ) ∼ (H \ K , T a ).
Of course, if (D, F ) is of type T a , then also (D\K, F ) ∼ (H \K , v a ) for some compact set K .Lemma 6.8.Let d ± = (a ± , θ ± ), (D, F ), B and t 0 be as in Proposition 6.6 and let T a be defined by (6.33).Let t 1 = max{t 0 , y a }.
Finally we note that because T a is conformal except in the vertical strip {z : 0 < Re z < 1}, we have (6.34)logarea supp(µ Ta ) ∩ ∆ < ∞.Since the inverse of the exponential function has only one singularity over 0 and ∞, while our hypothesis says that F −1 has infinitely many singularities over 0 and ∞, we deduce that the case m = 0 does not occur.Thus m ≥ 1.
Next we show that F −1 actually has infinitely many singularities over both 0 and ∞.To do so, we may assume without loss of generality that F −1 has infinitely many singularities over 0. Lemma 6.4 yields that F is unbounded in the region "between" two tracts over 0. Since all poles are real, this means that with at most two exceptions the region between two tracts over 0 must contain a tract over ∞.
To each asymptotic value a ∈ C * we associate a semi-open segment a = (a, a+εe iθ(a) ].Here ε > 0 is chosen so small that the disks D(a, 2ε) are disjoint and do not contain 0. The choice of θ(a) ∈ {0, ±π/2, π} will be fixed later.
For an asymptotic value a ∈ C * , there exists at least one unbounded component U of F −1 (D(a, 2ε)).For each such component U , the map F : U → D(a, 2ε) \ {a} is a universal cover.Hence U contains a component γ U of F −1 ( a ).Clearly γ U is a curve tending to ∞ in U .For each component U we fix such a curve γ U .So overall there are m such curves and they are disjoint.If U is symmetric, then we may choose γ U as a subset of R. Then θ(a) ∈ {0, π}.If symmetry interchanges two components, then we choose the corresponding angles and curves such that θ(a) = −θ(a) and γ U = γ U .
If F has a two-sided sequence of zeros and poles, then no such curve γ U can be contained in R. Hence the set of these curves splits into complex conjugate pairs.It follows that the number m must be even in this case.
Since the curves γ U tend to ∞, there is a cyclic order on the set of these curves.We enumerate them counterclockwise.This enumeration is independent of the choice of the angles θ(a).If there are infinitely many positive zeros and poles, we do it in such a way that the positive ray is between γ m and γ 1 .Let a j be the asymptotic value along γ j .By symmetry, we have a j = a m−j+1 and γ j = γ m−j+1 for 1 ≤ j ≤ m.We put γ 0 = γ m and a 0 = a m .Lemma 7.2.If F has infinitely many positive zeros and poles, then a 1 = a m so that a 1 = a m / ∈ R and m ≥ 2. If, in addition, F has infinitely many negative zeros and poles, then m is even, a m/2 = a m/2+1 and a m/2 = a m/2+1 / ∈ R.
Proof.Suppose that a 1 = a m so that a := a 1 = a m = a 1 ∈ R. Consider a symmetric cell decomposition C 0 of the sphere C with two vertices × = i and • = −i, three edges and three faces containing 0, ∞ and a, respectively; see Figure 4. We may assume that the edges do not intersect any of the disks D(a j , 2ε).Thus all these disks are contained in some face of C 0 .Let L 0 = F −1 (C 0 ).Since F is real and locally univalent, F is either always decreasing or always increasing between two adjacent poles.This implies that if x 0 is the smallest positive pole and if (x k ) denotes the sequence of all poles, zeros and a-points greater than or equal to x 0 , ordered such that x 0 < x 1 < x 2 < . . ., then x k is a pole if k is divisible by 3, and either x k is a zero if k ≡ 1 (mod 3) and an a-point if k ≡ 2 (mod 3), or vice versa.
Let X k be the face of L 0 containing x k and let Y k = F (X k ) be the corresponding face of C 0 .Suppose that F is not a covering from X k \ {x k } to Y k \ {F (x k )}.Then there exists j ∈ {1, . . ., m} such that γ j ⊂ X k and D(a j , 2ε) ⊂ Y k .This can happen for at most m values of k.Thus there exists K ∈ N such that F is a covering from This implies X k is a digon for all k ≥ K. Now X K shares an edge with X K+1 , and these two edges end at the same vertices.Next, X K+1 shares an edge with X K+2 , and again these two edges end at the same vertices.We conclude that the edges of X K , X K+1 and X K+2 end all at the same vertices.This is a contradiction, since only three edges can meet at a vertex.
The conclusion that a m/2 = a m/2 + 1 if F has infinitely many negative zeros and poles follows by considering F (−z) instead of F (z).
Let θ j = θ(a j ).Lemma 7.2 says that if F has infinitely many positive zeros and poles, then Im a 1 = − Im a m = 0.In order to apply Lemma 6.8, we choose θ 1 = − sign(Im a 1 )π/2 and θ m = −θ 1 .Similarly, if F has infinitely many negative zeros and poles, then we choose θ m/2 = − sign(Im a m/2 )π/2 and θ m/2+1 = −θ m/2 .If γ j is contained in the real axis, then, as already mentioned, we have θ j = 0 or θ j = π.For all other j the choice of θ j is irrelevant, but to be definite we choose θ j = 0 for these j.We put d j = (a j , θ j ).
We connect the endpoints of γ j to some point in R by curves σ j which are pairwise disjoint except for their common endpoint in R.Moreover, we assume that σ j intersects γ j only at its endpoint, and does not intersect any other γ k .Then, for 1 ≤ j ≤ m, there exists an unbounded domain G j whose boundary is formed by the curves γ j−1 , γ j , σ j−1 and σ j .
The proof of Theorem 1.3 is split into two parts: The first part (given in this section) is a purely topological statement, and the second part (in the next section) deals with the asymptotic behavior.
The topological statement is the following.Proof.Definition 6.1 says that if F has no zeros and poles in G j , then (G j , F ) is of type B d j ,d j+1 .This already proves (i ) and also shows that in case (ii ) the (G j , F ) are of the stated type if j = 0 and that in case (iii ) they are of the stated type if j = 0 and j = m/2.
To deal with (G 0 , F ) in cases (ii ) and (iii ), we consider a symmetric cell decomposition C 1 of the sphere with two vertices × and • on the real line, four edges and four faces containing 0, ∞, a := a 1 and a = a m , respectively, see the left part of We may assume that if a j is an asymptotic value with a j / ∈ {a, a}, then the disk D(a j , 2ε) is contained in one of the faces labeled 0 and ∞.Let L 1 = F −1 (C 1 ).Then the restriction of F to an unbounded face of L 1 labeled by a or a is a universal cover from this unbounded face to the corresponding face of C 1 punctured at a or a, respectively.
The same argument as in the proof of Lemma 7.2 shows that all sufficiently large positive poles and zeros of F must belong to digons of L 1 .Each such digon has both vertices on the positive ray.Conversely, every sufficiently large positive vertex belongs to two digons, one labeled 0 and one labeled ∞.
Thus there is a one-sided chain of the form × = • = × = • = × = . . .infinite in the positive direction, consisting of faces labeled 0 and ∞ alternatively.Consideration of the cyclic order of face labeling around vertices of these digons containing zeros and poles shows that immediately above and below this chain we must have ∞-gons labeled a and a; cf. the right part of Figure 5.
These ∞-gons must contain γ 1 and γ m , since there are no other asymptotic curves γ j with asymptotic values in C * between γ m and γ 1 in the sense of the counterclockwise cyclic order at ∞. Let D be the region consisting of these two ∞-gons and the closure of digons of chain.
Recall that T a 1 is defined in (6.33) by T a 1 = v a 1 • φ a 1 , with v a 1 given by (6.25) and a quasiconformal map φ a 1 .The cell decomposition v −1 a 1 (C 1 ) consists of a two-sided infinite chain of digons the form . . .= × = • = × = • = . . .and two ∞-gons.Removing the closure of the left part of this infinite chain, we obtain a region D with a cell decomposition combinatorially equivalent to the restriction of the cell decomposition L 1 on D. Therefore, the restriction of F to D is equivalent to the restriction of v a 1 to D , that is v a 1 = F • ψ, where ψ : D → D is a homeomorphism.Since F and v a 1 are symmetric, ψ can also be chosen to be symmetric.Moreover, since γ m = γ 1 and since F maps the curves γ 1 and γ m onto the segments (a 1 , a 1 + εe iθ 1 ] = (a, a − sign(Im a)i] and (a m , a m + εe iθm ] = (a, a + sign(Im a)i], we see that there exists k ∈ Z such that ψ −1 (γ 1 ) and ψ −1 (γ m ) are contained in the line {z : Re z = kπ}.There is no loss of generality to assume that they are on the imaginary axis.Now we extend the curves γ 1 and γ m until they hit some vertices on the boundary of their ∞-gons.This can be done in such a way that ψ −1 (γ 1 ) and ψ −1 (γ m ) are contained in the right half-plane.These extended curves cut from D a region D 0 which contains a positive ray.The restriction of F on D 0 is equivalent to the restriction of v a 1 onto the region ψ −1 (D 0 ), and hence the restriction of T a 1 onto φ −1 (ψ −1 (D 0 )).This yields that (G 0 , F ) is of type T a 1 .This completes the proof of (ii ) and also handles the case of (G 0 , F ) in case (iii ).
To complete the proof in case (iii ) we only have to note that considering F (−z) instead of F (z) corresponds to interchanging G 0 and G m/2 as well as a 1 and a m/2+1 .

Completion of the proof of Theorem 1.3: Gluing pieces together
Let (i ) − (iii ) be the cases considered in Theorem 7.3.Of course, these correspond to the cases (i) − (iii) of Theorem 1.3.We divide the plane into m sectors S 0 , . . ., S m−1 , enumerated counterclockwise and such that S 0 is bisected by the positive real axis.Let σ j be the opening angle of S j .We choose ρ and the opening angles σ j as follows: Case (i ): ρ = m and σ j = 2π/ρ for all j.Case (ii ): ρ = m − 1/2, σ 0 = π/ρ and σ j = 2π/ρ for 1 ≤ j ≤ m − 1. Case (iii ): ρ = m − 1, σ 0 = σ m/2 = π/ρ and σ j = 2π/ρ for all other j.Recall here that m is even in case (iii ) by Theorem 7.3.We call a sector S j large if σ j = 2π/ρ and small if σ j = π/ρ.If S j is a large sector, then there exists e j ∈ C with |e j | = 1 such that z → e j z ρ maps S j conformally onto Ω 0 = C \ [0, ∞).In fact, we have e j = −1 for all j in case (i ) and e j = −i for all j in case (ii ).In case (iii ) we have e j = −i for 1 ≤ j ≤ m/2 − 1 and e j = i for m/2 + 1 ≤ j ≤ m − 1.
If S 0 is a small sector, then z → z ρ maps S 0 onto the right half-plane H.If S m/2 is a small sector, which happens only in case (iii ), then z → −z ρ maps S 0 onto the right half-plane.Putting e 0 = 1 and e m/2 = −1 we see that if S j is a small sector and thus j ∈ {0, m/2}, then z → e j z ρ maps S j to H. Let p j : S j → C, p j (z) = e j z ρ .Then p j (S j ) = Ω 0 or p j (S j ) = H, depending on whether S j is large or small.Let d j = (a j , θ j ) be as in section 7 and let G 0 , . . ., G m−1 be the domains defined before Theorem 7.3.By Theorem 7.3, each element (G j , F ) is of one of two types.If it is of type B d j ,d j+1 , we choose the map B j and compact sets K j and K j according to Proposition 6.6 so that (Ω \ K j , B j ) ∼ (G j \ K j , F ). Otherwise there are compact sets K j and K j such that so that (H \ K j , T a j+1 ) ∼ (G j \ K j , F ).Note that our labeling of the sectors is such that We consider the map F 1 : m−1 j=0 S j → C, which for z ∈ S j of sufficiently large modulus is defined by Proposition 6.6 and Lemma 6.8 yield that, apart from some bounded set, the expressions defining F 1 match on the boundaries of the sectors.Thus there exists R > 0 such that (8.2) defines a quasiregular map F 1 : {z : |z| > R} → C.
By (8.1) there exist homeomorphisms φ j : p j (S j ) \ K j → G j \ K j such that (8.3).Hence the τ j can be glued together to yield compact sets K and K and a homeomorphism τ : On the other hand, as explained in § 4.1, the Uniformization Theorem yields that there exists 0 < R ≤ ∞, a homeomorphism φ 0 : C → D(0, R) and a meromorphic function Since F 0 is meromorphic and F 1 is quasiregular, we find that α is quasiconformal.Since a quasiconformal map distorts the modulus of an annulus only by a bounded factor, this implies that R = ∞.
The set where α is not conformal agrees with the set where F 1 is not meromorphic.By Lemma 4.5, (6.7) and (6.34) this set has finite logarithmic area.It thus follows from the Teichmüller-Wittich-Belinskii theorem (Lemma 4.4) that there exists a ∈ C * such that α(z) ∼ az as z → ∞.
It will be convenient to consider the inverse β = α −1 .With b = 1/a we then have (8.4) As F 0 and F 1 are symmetric, β is also symmetric.This implies that b is real.In fact, we may assume that b > 0.
It follows from the definition of the T a j and B j , (8.2), (8.4) and (8.5) that log m(r, F 0 ) = O(r ρ ) as r → ∞.Moreover, we find that if F 0 has infinitely many zeros and poles, then there exists a positive constant C such that In fact, we have C = b ρ /(2π) if the sequence of zeros and poles is one-sided and C = b ρ /π if it is two-sided.It follows from these equations and the lemma on the logarithmic derivative [22, Chapter 3, § 1] that F 0 /F 0 and hence E = F 0 /F 0 have order ρ.Moreover, it follows that λ(E) = ρ in cases (ii) and (iii).
To prove that E is of completely regular growth, we first consider a large sector S j .Let S j be a subsector of S j which is mapped to a subsector of the left half-plane under p j .By (6.9) we have log F 0 (z) = log B j (p j (β(z)))) ∼ cp j (β(z)) d exp(−p j (β(z))) as z → ∞, z ∈ S j and thus log log F 0 (z) ∼ −p j (β(z)) ∼ −e j (bz) ρ as z → ∞, z ∈ S j .
Replacing, without changing notation, S j by a smaller subsector we find that F 0 (z) F 0 (z) log F 0 (z) ∼ −e j ρb ρ z ρ−1 as z → ∞, z ∈ S j .
Let now S j be a large sector and let S j be a subsector which is mapped by p j to a subsector of the first quadrant {z : Re z > 0, Im z > 0}.By (6.8) we have F 0 (z) − a j = log B j (p j (β(z)))) − a j ∼ cp j (β(z)) d exp(−p j (β(z))) as z → ∞, z ∈ S j Similarly as above this yields (8.7) log(F 0 (z) − a j ) ∼ −p j (β(z)) ∼ −e j (bz) ρ as z → ∞, z ∈ S j and thus, passing to a smaller subsector, ∼ −e j ρb ρ z ρ−1 as z → ∞, z ∈ S j .
Thus log E(z) ∼ − log(F 0 (z) − a j ) and (8.7) yields that (8.6) holds again.The case that S j is mapped by p j to a subsector of the fourth quadrant {z : Re z > 0, Im z < 0} is analogous.Then the above equations hold with a j replaced by a j+1 , and again we obtain (8.6).
Next we consider the case that S j is a small sector.Again, let S j be a subsector which is mapped by p j to a subsector of the first quadrant.Then An analogous argument shows that if S j is mapped by p j to a subsector of the fourth quadrant, then (8.9) log E(z) ∼ ip j (β(z)) ∼ ie j (bz) ρ as z → ∞, z ∈ S j .
Suppose now that we are in case (ii).Then (8.6) holds for 1 ≤ j ≤ m − 1.If j = 0 and S 0 is a subsector of S 0 which is contained in the upper half-plane, then p j (S 0 ) is contained in the first quadrant and thus we have (8.8) with j = 0. Recalling that e 0 = 1 and e j = −i for all other j in case (ii), we find that if T is any closed subsector of the upper half-plane whose image under z → z ρ does not intersect the real or imaginary axis, then (8.10) log E(z) ∼ −ib ρ z ρ as z → ∞, z ∈ T.
Since E is symmetric, an analogous result holds for subsectors of the lower half-plane.Thus E is of completely regular growth on every ray except for finitely many.Since the set of rays of completely regular growth is closed [40, § III.1], E is of completely regular growth in the plane, with indicator as stated.
An analogous reasoning can be made in case (iii).In this case a subsector of S m which is contained in the upper half-plane is mapped to the fourth quadrant.Thus we have to use (8.9) instead of (8.8) if j = m/2.But since e m = −1 we again find that (8.10) holds for any subsector T of the upper half-plane whose image under z → z ρ does not intersect the real or imaginary axis.As before we can conclude that E has completely regular growth, with indicator as stated.
Finally, to prove that A has completely regular growth, we note that it follows from (8.10) that if T is a closed sector containing no zeros of E, then there exists a constant c such that (8.11) −2 Since E has completely regular growth this implies together with (1.4) that A has completely regular growth, with indicator given by (1.7).
Remark 8.1.It follows from (8.11) that if E has infinitely many positive zeros, then (8.12) A(z) ∼ 1 4 c z 2ρ−2 as z → ∞, z ∈ T, for any closed subsector T of S 0 \R.Lemma 6.4, applied to A(z)/z 2ρ−2 , shows that (8.12) in fact holds for any closed subsector T of S 0 .An analogous result holds if E has infinitely many negative zeros.Thus we actually have a much more precise description of the asymptotics of A than given by (1.7):In the sectors corresponding to the intervals where h A = 0 we have (8.12).
In particular is follows from (8.12) that A is non-constant.But this can also be deduced directly from the hypothesis that F −1 has infinitely many singularities over 0 or ∞.Remark 8.2.Suppose that E has infinitely many positive zeros.Since between two positive zeros of a solution w of (1.1) there is positive local maximum or a negative local minimum of w, we deduce from (1.1) that c > 0 in (8.12).This implies that A (x) > 0 for all large positive x.It follows (see [10] or [25, Chapter XIV, Part I, Theorem 3.1]) that all solutions of (1.1) are bounded on the positive real axis.In particular, E is bounded there.Alternatively, this can be obtained from F 0 (z) = T a 1 (p j (β(z)), which holds for z of sufficiently large modulus in any subsector of S 0 .Again an analogous result holds if E has infinitely many negative zeros.
Since E has only finitely many non-real critical points by Lemma 3.6, this yields that the set of critical values of E is bounded.Moreover, by the Denjoy-Carleman-Ahlfors theorem [22, Chapter 5, § 1], E has only finitely many asymptotic values.We conclude that E is in the class B consisting of all entire functions for which the set of critical and (finite) asymptotic values is bounded.It plays an important role in value distribution and holomorphic dynamics [52], as does the Speiser class.Lemma 9.2.Let F : D → C be a normal meromorphic function.Suppose that there exist a ∈ C and a curve γ ending at point P ∈ ∂D such that F (z) → a as z → P , z ∈ γ.Then f has the angular limit a at P .Proof of Theorem 1.4.Let w 1 , w 2 , w 3 be pairwise linearly independent solutions of (1.1) with only real zeros.Without loss of generality we may assume that w 3 = w 1 − w 2 , since otherwise we can replace w 1 and w 2 by suitable multiples.As before we put F = w 2 /w 1 .Then F is a locally univalent meromorphic function which has only real zeros, 1-points and poles.
In view of (1.2) we have to show that the Schwarzian derivative of F is constant.This is the case if F has the form (9.1).Using Lemma 9.1 we may thus assume that F maps R to a circle C.
The Schwarzian derivative of a linear-fractional transformation is 0. So we may assume that F is not a linear-fractional transformation.Then the inverse of F has a singularity.Since F is locally univalent, this means that F has an asymptotic value a.Let γ be an asymptotic path for a.Then γ is an asymptotic path for a * , the point symmetric to a with respect to the circle C. If γ crosses the real axis infinitely often, then a = a * .In this case we can build from γ and γ an asymptotic path which is contained in the upper half-plane H + .If a = a * is non-real, then (the tail of) one of the curves γ or γ is in H + anyway, and we may assume without loss of generality that this holds for γ.Thus F has the asymptotic value a with an asymptotic path in the upper half-plane.
As F omits the values 0, 1 and ∞ in the upper half-plane, F is normal there.Lemma 9.2 yields that F (z) → a as |z| → ∞, ε < arg z < π − ε.In particular, F has only one asymptotic value which has an asymptotic path in the upper half-plane.Overall we see that F has at most two asymptotic values, namely a and a * .Thus the inverse of F has at most two singularities.In fact, this also shows that a ∈ C so that

( a )
If the zeros of E form an infinite one-sided sequence, then A is non-constant, m ≥ 2 and λ(E) ≥ m − 1/2.(b) If the zeros of E form an infinite two-sided sequence and A is non-constant, then m ≥ 4 and λ(E) ≥ m − 1.
entire and has the following properties: (i) If F has only finitely many zeros and poles, then m ≥ 1 and ρ(E) = m.(ii) If the zeros and poles of F form an infinite one-sided sequence, then m ≥ 2 and λ(E) = ρ(E) = m − 1/2.(iii) If the zeros and poles of F form an infinite two-sided sequence, then m is even, m ≥ 4 and λ(E) = ρ(E) = m − 1. (iv) The functions E and A in (1.4) have the same order ρ = ρ(E), and they are of completely regular growth in the sense of Levin-Pfluger.

Lemma 3 . 7 .
Let f : [a, b] → R be C 2 with the following properties.(a) f (a) = f (b) = 0. (b) |f (a)| = |f (b)| ≥ 1. (c) f has a unique zero c in (a, b).(d) f has at most one zero in [a, c] and at most one in [c, b].
defines a harmonic function v : C → R. Next there exists an entire function w with Re w(0) = 0 such that v = Im w.It follows from (3.12) that w(0) = 0 and Im w(z) ≤ |z| σ for z ∈ C.

4 .
and |z| is large enough.Together with (1.4) we see that |A(z)| = O( (|z|) 2 ) and hence |A(z)| = O(|z| ε ) as |z| → ∞, ε ≤ | arg z| ≤ π − ε.Since the order of A is finite, for sufficiently small ε > 0 the Phragmén-Lindelöf principle [40, § I.14]shows that the last estimate also holds for | arg z| ≤ ε and | arg z − π| ≤ ε.It follows that A is constant.A similar argument can be made in case (a).Remark 3.5.It follows from the proof of Theorem 1.2 that the Nevanlinna deficiency δ(0, E) is positive.In fact,(3.16) and (3.17) yield that δ(0, E) = (n − 1)/n.Preliminaries for the proof of Theorem 1.3 is a covering.For a linear-fractional transformation, this holds with A = f (∞), and to avoid trivialities, we assume that f has at least two singular values (critical or asymptotic values), so the set A contains at least two points.For such an f ∈ S, consider a cell decomposition C of C and a finite set A of its singular values.Let L f = f −1 (C) be the preimage of C. Then L f is a labeled cell decomposition of C satisfying the conditions stated in § 4.3.This labeled cell decomposition L f is called the line complex of f corresponding to C. It has the following properties:(a) Its 1-skeleton is a bipartite graph embedded in the plane.(b) The cyclic order of face labels around each ×-vertex is the same: it coincides with the cyclic order of face labels around the ×-vertex of C. The cyclic order of face labels around an •-vertex is opposite.For fixed A and C, any cell decomposition of C with these two properties arises from some topologically holomorphic map of class S.It follows from Proposition 4.2 that any two functions f and g of class S with the same labeled cell decomposition L f = L g (and same A and C) are equivalent: f = g • φ, where φ : C → C is a homeomorphism.

5. 1 . 0 e −t m − 1 t dt = exp 1 0e −t m − 1 t dt + z 1 e
Examples without zeros or with one zero.For case (i) in Theorem 1.3, functions with the required properties are easily given.Given m ∈ N, g m (z) = exp z 0 e ζ m dζ is a real local homeomorphism with m singularities over C * which has no zeros or poles while (5.1) h m (z) = z exp z −t m t dt is a real local homeomorphism with m singularities over C * and one zero at the origin.

2 .
The second type B d 1 ,d 2

Figure 2 .
Figure 2. The cell decomposition for the trees in Figures 1 and 3.

Figure 3 .
Figure 3.The tree for m = 6.(Since the tree is symmetric, only its upper part is shown.Unlabeled faces are asymptotic tracts over 0 or ∞.)

6. 1 .Definition 6 . 1 .
Elements without zeros and poles.The domains where F maps to C * are covered by the following definition.Let D be an unbounded simply-connected domain bounded by a curve γ : R → C, with γ(t) → ∞ as t → ±∞, and let F : D → C * be a local homeomorphism.Suppose that if c ∈ C * , then there exists δ > 0 such that F −1 (D(c, δ)) has no unbounded connected component whose closure is in D. Let d ± = (a ± , θ ± ), where a ± ∈ C * and θ ± ∈ (−π, π], with θ + = θ − if a + = a − .If (D, F ) has the oriented asymptotic values d ± at ±∞, then (D, F ) is called of type B d + ,d − .

7 .
Beginning of the proof of Theorem 1.3: Cutting into pieces Let F be a symmetric local homeomorphism of class S with a finite number m of singularities of F −1 over C * .Suppose first that m = 0. Then F : C → C * is a covering.This leads easily to the following result.Lemma 7.1.Let F : C → C be a local homeomorphism such that the inverse F −1 has no singularities over points in C * .Then (C, F ) ∼ (C, exp).

Figure 4 .
Figure 4.The symmetric cell decomposition C 0 in the proof of Lemma 7.2.

Theorem 7 . 3 .
Let F : C → C be a symmetric local homeomorphism such that the inverse F −1 has a finite, non-zero number m of singularities over points in C * .Then:(i ) If F has only finitely many zeros and poles, then (G j , F ) is of type B d j ,d j+1 for all j.(ii ) If F has infinitely many zeros and poles, all of them positive, then m ≥ 2 and(G 0 , F ) is of type T a 1 while (G j , F ) is of type B d j ,d j+1 for 1 ≤ j ≤ m − 1. (iii )If F has a two-sided sequence of zeros and poles, all of them real, then m is even and m ≥ 2. Moreover,(G 0 , F ) is of type T a 1 and (G m/2 , F ) is of type T a m/2+1while (G j , F ) is of type B d j ,d j+1 for all other j.

9 .
Proof of Theorem 1.4 We will use the following result [6, Theorem 1].Lemma 9.1.Let F be a meromorphic function such that the preimage of three points belongs to the real line.Then F maps the real line into a circle, unless (9.1)F (z) = L 1 − e i(a 1 z−b 1 ) 1 − e i(a 2 z−b 2 ) ,where L is a linear-fractional transformation and a j , b j ∈ R. Let D be the unit disk.A meromorphic function F : D → C is called normal if the family {F • S : S ∈ Aut(D)} is normal, where Aut(D) denotes the set of biholomorphic maps from D to D. The following result is due to Lehto and Virtanen [38, Theorem 2].