---
abstract: 'The Eremenko-Lyubich class of [[transcendental entire function]{}]{}s with a bounded set of singular values has been much studied. We give a new characterisation of this class of functions. We also give a new result regarding direct singularities which are not logarithmic.'
address: |
    Department of Mathematics and Statistics\
    The Open University\
    Walton Hall\
    Milton Keynes MK7 6AA\
    UK
author:
- 'D. J. Sixsmith'
bibliography:
- 'Main.bib'
title: 'A new characterisation of the Eremenko-Lyubich class'
---

Introduction
============

Throughout this paper we assume that $f$ is a [[transcendental entire function]{}]{}. We say that $f$ belongs to the Eremenko-Lyubich class, $\mathcal{B}$, if the set of finite critical and asymptotic values of $f$ is bounded. This set coincides with the set of singular values of the inverse function $f^{-1}$. The class $\mathcal{B}$ was introduced to complex dynamics in [@MR1196102], and has been widely studied; see, for example, the papers [@MR2346947], [@MR2570071], [@MR1610785] and [@MR2753600] on the structure of the escaping set for functions in this class; [@MR2480096], [@MR2587450] and [@MR1357062] on the dimensions of the Julia set and the escaping set; and [@Bish1] for an example of a function in this class with a wandering domain. Papers studying the value distribution of functions in this class include [@MR2081677], [@MR1601855] and [@MR2478274].

An important property of functions in the Eremenko-Lyubich class is that they are expanding, in the following sense. Define $D_R = \{ z : |f(z)| > R \}$, for $R>0$. If $f\in\mathcal{B}$, then it follows easily from [@MR1196102 Lemma 1] that there is a constant $R_0>0$ such that $$\label{expeq}
\left|z\frac{f'(z)}{f(z)}\right| \geq \frac{1}{4\pi}(\log|f(z)| - \log R_0), {\quad\text{for }}z \in D_{R_0}.$$ This property has many applications in complex dynamics and value distribution theory; for example, it was used in [@MR1196102] to show that functions in the Eremenko-Lyubich class cannot have escaping Fatou components.

Define $$\label{def_of_s}
\eta_f = \lim_{R\rightarrow\infty} \inf_{z \in D_R} \left|z \frac{f'(z)}{f(z)}\right|.$$ It follows from (\[expeq\]) that if $f$ is a [[transcendental entire function]{}]{} in the Eremenko-Lyubich class, then $\eta_f~=~\infty$. The main result of this paper is the following, which shows that this property has a strong converse.

\[theo1\] Suppose that $f$ is a [[transcendental entire function]{}]{}. Then, either $\eta_f = \infty$ and $f\in\mathcal{B}$, or $\eta_f = 0$ and $f\notin\mathcal{B}$.

It is clear that if $f$ has an unbounded set of critical values, then $\eta_f=0$. Thus the proof of Theorem \[theo1\] requires detailed analysis of the behaviour of functions with an unbounded set of asymptotic values. Since every asymptotic value of $f$ gives rise to a [[transcendental singularity]{}]{} of $f^{-1}$, we need a number of results on singularities of the inverse function. In particular we require the following result on the density of [[transcendental singularities]{}]{} of a certain type, which may be of independent interest. Definitions of terms used in the statement of this theorem are given in Section \[S1\].

\[theo2\] Suppose that $f$ is a [[transcendental entire function]{}]{}, with a [[direct non-logarithmic singularity]{}]{} with projection $a\in\widehat{\mathbb{C}}$. Then at least one of the following holds:

(i) $a$ is the limit of critical values of $f$;

(ii) every neighbourhood of this singularity contains a neighbourhood of another [[transcendental singularity]{}]{} of $f^{-1}$ that is either indirect or logarithmic, and whose projection is different from $a$.

We observe that Theorem \[theo2\] is complementary to the following result of Bergweiler and Eremenko [@MR2507243 Theorem 5], which has the same hypothesis although in this result the projection of the [[transcendental singularity]{}]{} must be finite.

\[OtherBergErem\] Suppose that $f$ is a [[transcendental entire function]{}]{}, with a [[direct non-logarithmic singularity]{}]{} with projection $a\iscomplex$. Then every neighbourhood of this singularity is also a neighbourhood of other [[direct singularities]{}]{} of $f^{-1}$ with projection $a$.

Taken together, these results show that if $a\iscomplex$ is the projection of a [[direct non-logarithmic singularity]{}]{} and is not the limit of critical values, then there is an infinite number of singularities both over $a$ and over points arbitrarily close to $a$.\
We mention two examples of [[transcendental entire function]{}]{}s with [[direct non-logarithmic singularities]{}]{} which illustrate some of the possibilities described above.

Heins [@MR0094457 p.435] gave the example $f_1(z) = e^z \sin(e^z)$, which has precisely one [[direct non-logarithmic singularity]{}]{} over $\infty$. Since the set of critical values of $f_1$ is unbounded, case $(i)$ of Theorem \[theo2\] holds for this function. This example also shows that Theorem \[OtherBergErem\] cannot be strengthened to $a\in\widehat{\mathbb{C}}$.

Herring [@MR1642181] gave the example $f_2(z) = \int_0^z \exp(-e^t) \ dt$. This function has no critical points. It follows from results in [@MR1642181] that $f_2$ has a [[direct non-logarithmic singularity]{}]{} over $\infty$, every neighbourhood of which contains a left half-plane. It also follows that within each set $$A_k = \{ z : \operatorname{Re}(z) > 0, |\operatorname{Im}(z) - 2k\pi|\leq \pi/2\}, {\quad\text{for }}k\in\mathbb{Z},$$ there is a neighbourhood of a [[logarithmic singularity]{}]{} with projection $$\alpha_k = \alpha + 2k\pi i, \text{ where } \alpha\iscomplex \text{ is constant}.$$ Moreover, each neighbourhood of the [[direct non-logarithmic singularity]{}]{} over $\infty$ contains neighbourhoods of these [[logarithmic singularities]{}]{}. Hence case $(ii)$ of Theorem \[theo2\] holds for $f_2$.

The structure of this paper is as follows. In Section \[S1\] we give details of Iversen’s classification of singularities. We then prove Theorem \[theo2\] in Section \[S1A\]. Finally, in Section \[S2\], we use Theorem \[theo2\] to prove Theorem \[theo1\].

Singularities of the inverse function {#S1}
=====================================

We write $\widehat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}$, and use $B(w, r)$ to refer to the open disc around the point $w\iscomplex$, of radius $r$. We also write $B(\infty, r) = \{~z~:~|z|~>~r\}$. We write $\disc$ for the unit disc $B(0, 1)$, and $\disc^*$ for the punctured unit disc $B(0, 1)\backslash\{0\}$. We write $\hplane$ for the left half-plane $\{z : \operatorname{Re}(z) < 0\}$.

We recall Iversen’s classification of singularities; see, for example, [@MR1344897], [@MR2507243], and [@Iversen]. Suppose that $f$ is a [[transcendental entire function]{}]{}, and suppose that $a\in\widehat{\mathbb{C}}$. For each $r > 0$, we can choose a component $U(r)$ of $f^{-1}(B(a, r))$ so that $r_1 < r_2$ implies that $U(r_1) \subset U(r_2)$. Then we have two possibilities:

(a) $\cap_{r>0} U(r)$ consists of a single point $w$, say, or

(b) $\cap_{r>0} U(r) = \emptyset$.

In the first case, if $f'(w) = 0$, then $w$ is a *critical point of $f$, $a$ is a *critical value of $f$, and we say that the singularity is *algebraic.***

In the second case we say that the choice $r \mapsto U(r)$ defines a *[[transcendental singularity]{}]{} of $f^{-1}$, and we say that $a$ is the *projection of the [[transcendental singularity]{}]{} or equivalently that the [[transcendental singularity]{}]{} is *over $a$. Any of the sets $U(r)$ is called a *neighbourhood of the [[transcendental singularity]{}]{}.****

We say that a [[transcendental singularity]{}]{} over a point $a$ is *direct if there exists $r > 0$ such that $f(z) \ne a$, for $z \in U(r)$. Otherwise we call the [[transcendental singularity]{}]{} *indirect. We call a direct [[transcendental singularity]{}]{} over a point $a$ *logarithmic if, for some $r>0$, the restriction $f : U(r) \rightarrow B(a, r)\backslash\{a\}$ is a universal covering. If a [[transcendental singularity]{}]{} is direct but not logarithmic, we use the term *direct non-logarithmic.****

We call a curve $\Gamma: (0, 1) \rightarrow \mathbb{C}$ an *asymptotic curve with *asymptotic value $a$ if, as $t\rightarrow 1$, we have both $\Gamma(t)\rightarrow\infty$ and $f(\Gamma(t))\rightarrow a$. Given a [[transcendental singularity]{}]{} over a point $a$ it is possible to construct an asymptotic curve with asymptotic value $a$, and *vice versa; see [@MR1344897 p.2] for details.***

[[direct non-logarithmic singularities]{}]{} {#S1A}
============================================

In this section we prove Theorem \[theo2\]. We need the following [@MR0094457 Theorem $4'$].

\[Heins\] Suppose that $f$ is a [[transcendental entire function]{}]{}, $D\subset\mathbb{C}$ is a domain, and $W$ is a component of $f^{-1}(D)$. Then either $f_W$, the restriction of $f$ to $W$, has finite constant valence in $D$, or else there is at most one point of $D$ at which the valence of $f_W$ is finite.

Here the *valence of a point $a\in D$ is the number of solutions of $f(z) = a$, for $z \in W$. It follows from Theorem \[Heins\] that there cannot be two distinct points $a, a' \in D$ such that $f(z) \in \{a, a'\}$ has no solutions, for $z \in W$.*

We also need the following result, and two straightforward corollaries of it. This seems to be well known, and follows quickly from results such as [@MR1326604 Example 4.2]. See also [@MR2757285 Theorem 6.1.1] for a detailed proof.

\[Tcovering\] Suppose that $W\subset\mathbb{C}$ is a domain, and $g: W \to \disc^*$ is an unbranched covering map. Then exactly one of the following holds:

(i) there exists a conformal map $\phi: W \to \hplane$ such that $g = \exp \circ \ \phi$;

(ii) there exists a conformal map $\phi: W \to \disc^*$ such that $g = (\phi)^m$, for some $m\isnatural$.

The first corollary is similar to [@MR2757285 Theorem 6.2.2], and differs from that result in that it specifies the location of the neighbourhoods of the singularities, which is necessary for the proof of Theorem \[theo2\]. We give a proof for completeness.

\[limitcorollary\] Suppose that $f$ is a [[transcendental entire function]{}]{} with a [[transcendental singularity]{}]{} which is not logarithmic, over a point $a\in\widehat{\mathbb{C}}$. Then at least one of the following holds:

(i) $a$ is the limit of critical values of $f$;

(ii) every neighbourhood of this singularity contains a neighbourhood of another [[transcendental singularity]{}]{} of $f^{-1}$ whose projection is different from $a$.

Suppose that, contrary to the conclusion of the corollary, we can choose a sufficiently small $r>0$ such that there are no critical points of $f$ in $$W=U(r)\backslash\{z:f(z)=a\},$$ and all [[transcendental singularities]{}]{} of $f^{-1}$ with a neighbourhood contained in $U(r)$ have projection $a$. The restriction of $f$ to $W$ is, therefore, an unbranched covering of $B(a, r)\backslash\{a\}$.

Let $h$ be a conformal map from $B(a, r)\backslash\{a\}$ to $\disc^*$. We apply Theorem \[Tcovering\] with $g = h \circ f$. If case $(i)$ of the theorem holds, then $W$ is simply connected, and the singularity is logarithmic, which is a contradiction. If case $(ii)$ of the theorem holds, then the conformal mapping $\phi$ has a punctured disc in the Riemann sphere as its domain, and at the puncture $\phi$ has a removable singularity. Hence, since $f=h^{-1} \circ (\phi)^m$, the singularity is algebraic; this is also a contradiction.

The second corollary of Theorem \[Tcovering\] is similarly straightforward, and we omit the proof.

\[dltscorollary\] Suppose that $f$ is a [[transcendental entire function]{}]{} with a [[logarithmic singularity]{}]{} over a point $a\in\widehat{\mathbb{C}}$. Then there exist a neighbourhood of the singularity, $W=U(r)$, and conformal maps $\phi: W \to \hplane$ and $h : B(a,r) \to \disc^*$ such that $h \circ f = \exp \circ \ \phi$.

We now prove Theorem \[theo2\].

Suppose that $f$ has a [[direct non-logarithmic singularity]{}]{} over a point $a~\in~\widehat{\mathbb{C}}$, and that $a$ is not the limit of critical values of $f$. The existence of [[transcendental singularities]{}]{}, over points other than $a$, in any neighbourhood of this [[direct non-logarithmic singularity]{}]{} follows from Corollary \[limitcorollary\]; we need to show that in any neighbourhood of this singularity there are singularities over points other than $a$, which are either logarithmic or indirect.

The structure of the proof is as follows. We assume the contrary, and construct a sequence of [[direct non-logarithmic singularities]{}]{} the projections of which have a limit. We show that this limit is itself the projection of a [[direct non-logarithmic singularity]{}]{}, and use the comment after Theorem \[Heins\] to obtain a contradiction. Figure 1 illustrates the points and sets constructed.

![The construction used in the proof of Theorem \[theo2\]](picture){width="12cm" height="9cm"}

Let $r_0>0$ be such that there are no critical values of $f$ in $B(a, r_0)$ and also, since the [[transcendental singularity]{}]{} is assumed to be direct, such that $f(z) \ne a$, for $z$ in the neighbourhood $U(r_0)$. Suppose also that $r_0$ is sufficiently small that all [[transcendental singularities]{}]{}, over points other than $a$ and with a neighbourhood contained in $U(r_0)$, are direct non-logarithmic.

Let $(R_n)$ be any increasing sequence of positive real numbers such that $R_n\rightarrow\infty$ as $n\rightarrow\infty$. We construct a sequence of neighbourhoods of [[direct non-logarithmic singularities]{}]{} $W_n(r_n)$, with projection $a_n$ say, such that, for $n\geq 0$,

-   $W_{n+1}(r_{n+1}) \subset W_n(r_n) \subset U(r_0)$;

-   $W_{n+1}(r_{n+1}) \cap B(0, R_{n+1}) = \emptyset$;

-   $\overline{B(a_{n+1}, r_{n+1})} \subset B(a_n, r_n)$;

-   $a_{n+1} \ne a$;

-   the equation $f(z) = a_n$ has no solutions for $z \in W_n(r_n)$;

-   $r_n \rightarrow 0$ as $n\rightarrow\infty$.

We set $W_0(r_0)=U(r_0)$, and then construct this sequence inductively. By assumption, $W_n(r_n)$ is a neighbourhood of a [[direct non-logarithmic singularity]{}]{}, so we can use Corollary \[limitcorollary\] to choose a [[transcendental singularity]{}]{} with projection $a_{n+1}$ say and with neighbourhoods $W_{n+1}(r), \ r>0,$ such that $W_{n+1}(r_{n+1}') \subset W_n(r_n)$, for some $r_{n+1}'>0$, and such that $0 < |a_{n+1} - a_n| < r_n/2$.

Next, we choose $r_{n+1}''>0$ such that $W_{n+1}(r_{n+1}'') \cap B(0, R_{n+1}) = \emptyset$. By assumption $W_{n+1}(r_{n+1}')$ is a neighbourhood of a [[direct singularity]{}]{} with projection $a_{n+1}$. Hence, there exists $r_{n+1}$ with $$0 < r_{n+1} < \min\{r_{n+1}', r_{n+1}'', |a_{n+1} - a_n|/4\}$$ such that $f(z) = a_{n+1}$ has no solutions for $z \in W_{n+1}(r_{n+1})$. Finally, both $\overline{B(a_{n+1}, r_{n+1})} \subset B(a_n, r_n)$ and $a_{n+1}\ne a$, by the choice of $a_{n+1}$ and $r_{n+1}$. This completes the construction.

Let $a' = \lim_{n\rightarrow\infty} a_n$, which exists by our choice of $r_n$. Note that, by construction, $a' \ne a$. Let $\Gamma$ be a curve produced inductively by joining a point in $W_n(r_n)$ to a point in $W_{n+1}(r_{n+1})$ using a curve lying in $W_n(r_n)$. By construction, $\Gamma$ is an asymptotic curve with asymptotic value $a'$. Hence $a'$ is the projection of a [[transcendental singularity]{}]{} of $f^{-1}$ which, by assumption, is direct non-logarithmic.

Choose $r>0$ sufficiently small that $D=B(a', r) \subset B(a, r_0)$, and such that $f(z) = a'$ has no solutions in $W$, where $W$ is the component of $f^{-1}(D)$ which has unbounded intersection with $\Gamma$.

Now, by construction, $a'\in B(a_n, r_n)$, for $n\isnatural$, and so there is an $N>0$ such that $a'\in B(a_N, r_N) \subset D$ and also $a' \ne a_N$. Note that $W_N(r_N) \subset W$, since the intersection of $\Gamma$ and $W_N(r_N)$ is unbounded. Then $a'$ and $a_N$ are two distinct points in $B(a_N, r_N)$ such that $f(z) \in \{a', a_N\}$ has no solutions in $W_N(r_N)$, which is contrary to Theorem \[Heins\].

The Eremenko-Lyubich Class {#S2}
==========================

In this section we prove Theorem \[theo1\]. We need the following, [@MR1230383 Theorem I.2.2].

\[Lindelof\] Suppose that $W \subset \widehat{\mathbb{C}}$ is simply connected and $\partial W$ has more than one point. Suppose that $\psi$ maps $W$ conformally to $\disc$. Let $\Gamma$ be a Jordan arc in $W$ with endpoint $z_0 \in \partial W$. Then the curve $\psi(\Gamma)$ terminates in a point $s_0\in \partial \disc$, and $\psi^{-1}(s) \rightarrow z_0$ as $s \rightarrow s_0$ inside any Stolz angle at $s_0$.

Here a *Stolz angle at $s_0\in \partial \disc$ is a set of the form; $$\{s\in\disc : | \arg(1 - \overline{s_0} s)| < \alpha, |s - s_0| < d \}, {\quad\text{for }}0 < \alpha < \pi/2, \ d < 2\cos\alpha.$$*

We also need the following result, which is a version of [@MR1344897 Theorem 1] that includes some assertions that appear only in the proof of that result; see also, [@MR2757285 Theorem 6.2.3].

\[BergErem\] Suppose that $f$ is a [[transcendental entire function]{}]{} with an [[indirect singularity]{}]{} with projection $a\in\widehat{\mathbb{C}}$. Suppose that $a$ is not the limit of critical values of $f$. Then there exists a sequence of asymptotic values $(a_n)$, which converge to $a$, a sequence of disjoint unbounded simply connected domains $(U_n)$ such that $D_n~=~f(U_n)$ is a disc with $a_n\in\partial D_n$, and a sequence of asymptotic curves $(\Gamma_n)$ such that $\Gamma_n \subset U_n$, $f(\Gamma_n)$ is a radius of $D_n$ ending at $a_n$, and $f$ is univalent in $U_n$.

Finally, we need the following lemma.

\[lem1\] Let $f$ be a [[transcendental entire function]{}]{}. Suppose that for every $R~>~0$ there exist $r>0$, $a_0\iscomplex$ with $|a_0|>R$, an asymptotic curve $\Gamma'$ with asymptotic value $a_0$, $W$ a simply connected neighbourhood of $\Gamma'$, and an analytic map $\phi$, univalent on $W$, such that $\phi(\Gamma')$ is an interval $(-\infty,x_0)$, and $$\label{newfeq}
f(z) = re^{\phi(z)} + a_0, {\quad\text{for }}z\in W.$$ Then $\eta_f = 0$.

Suppose that $\eta_f \neq 0$. Then there exist $\epsilon, R>0$ such that $$\label{sisnotzero}
\left|z\frac{f'(z)}{f(z)}\right| > \epsilon, {\quad\text{for }}|f(z)| > R.$$ Choose $a_0$ such that $|a_0|> 2R$, let $h = \phi^{-1}$ and put $t=\phi(z)$. Then, as $z\rightarrow\infty$ along $\Gamma'$, by (\[newfeq\]) and (\[sisnotzero\]), $$\begin{aligned}
\epsilon &< \left| z \frac{\phi'(z) r e^{\phi(z)}}{re^{\phi(z)} + a_0} \right| 
          = \left| h(t) \frac{\phi'(h(t)) re^{t}}{re^{t} + a_0} \right| 
          = \left| \frac{h(t) re^{t}}{h'(t) (re^{t} + a_0)} \right| 
          \sim \left| \frac{h(t) re^{t}}{h'(t) a_0} \right|.\end{aligned}$$ Hence, for sufficiently large negative values of $t$, $$\label{ee}
\left| \frac{h'(t)}{h(t)} \right| < \frac{2re^{t}}{\epsilon |a_0|}.$$ Without loss of generality we can assume that (\[ee\]) applies for all $t\in (-\infty,x_0)$. Since $W$ is simply connected, and since we can assume that $0\notin W$, we can define a branch of the logarithm, $L$, in $W$. Then, by (\[ee\]), $$\label{ez}
\left| \frac{d}{dt} L(h(t)) \right| < \frac{2re^{t}}{\epsilon |a_0|}.$$

We set $\zeta = L(h(t))$ and integrate (\[ez\]), to obtain $$\label{ineq}
\frac{2r}{\epsilon |a_0|} \int_{-\infty}^{x_0} e^{t} \ dt > \int_{-\infty}^{x_0} \left|\frac{d}{dt} L(h(t))\right| \ dt  > \left| \int_{-\infty}^{x_0} \frac{d}{dt} L(h(t)) \ dt \right| = \left| \int_{L(\Gamma')} d\zeta \right|.$$

Now, $L(\Gamma')$ is an unbounded curve, and so the right-hand side of (\[ineq\]) is unbounded. However, the left-hand side of (\[ineq\]) is bounded. This contradiction completes the proof.

We now prove Theorem \[theo1\].

As mentioned in the introduction, it is clear that if $f\in\mathcal{B}$ then $\eta_f=\infty$. Suppose, then, that $\eta_f \ne 0$. It is immediate from (\[def\_of\_s\]) that the set of critical values of $f$ is bounded. To complete the proof, we show that $f$ cannot have an unbounded set of finite asymptotic values, and so $f\in\mathcal{B}$, and hence $\eta_f = \infty$. To achieve this we show first that $f$ cannot have an unbounded set of projections of [[logarithmic singularities]{}]{}. We then show that $f$ cannot have an unbounded set of projections of [[indirect singularities]{}]{}. Finally, we show that $f$ cannot have an unbounded set of projections of [[direct non-logarithmic singularities]{}]{}.

Our first claim then is that $f$ cannot have an unbounded set of projections of [[logarithmic singularities]{}]{}. Suppose that, for every $R>0$, $f$ has a [[logarithmic singularity]{}]{} with projection $a_0\iscomplex$, such that $|a_0|~>~R$. Noting that $a_0$ is finite, we apply Corollary \[dltscorollary\] to obtain a simply connected neighbourhood, $W=U(r)$, of the singularity, and a conformal map $\phi: W \to \hplane$ such that (\[newfeq\]) holds for some $r>0$. Let $\Gamma$ be an asymptotic curve in $W$ associated with the [[logarithmic singularity]{}]{}.

Put $t=\phi(z)$ and let $$s = \sigma(t) = \frac{1-t}{1+t}.$$ Then $\psi = \sigma \circ \phi$ is a conformal mapping of $W$ to $\disc$. Moreover $\psi(\Gamma)$ is a curve in $\disc$ tending to $-1$.

We now construct another curve to $\infty$ in $W$ which satisfies the hypotheses of Lemma \[lem1\]. Let $\Gamma' = \phi^{-1}((-\infty, 0))$. Then $\gamma = \psi(\Gamma')$ is a curve in $\disc$ tending to $-1$ within a Stolz angle. By Theorem \[Lindelof\], $\psi^{-1}(s) \rightarrow \infty$ as $s\rightarrow -1$ along $\gamma$, and so $\Gamma'$ is an asymptotic curve. Moreover, $re^{t} + a_0 \rightarrow a_0$ as $t \rightarrow -\infty$ along $\phi(\Gamma')$, and so $\Gamma'$ has asymptotic value $a_0$. A contradiction follow by Lemma \[lem1\], since we have assumed that $|a_0|>R$. This establishes our initial claim.

We next show that $f$ cannot have an unbounded set of projections of [[indirect singularities]{}]{}. Suppose that, for every $R>0$, $f$ has an [[indirect singularity]{}]{} with projection $a\iscomplex$, such that $|a|>2R$. By Theorem \[BergErem\], $f$ has an asymptotic value $a_0$, with $|a_0|>R$, an asymptotic curve $\Gamma'$ associated with $a_0$, and an unbounded simply connected domain $W$ containing $\Gamma'$ such that $f$ is univalent in $W$. Moreover, $f(W)$ is a disc, $D$, with $a_0 \in \partial D$, and $f(\Gamma')$ is a radius in $D$ ending at $a_0$.

Without loss of generality, by composing with a rotation if necessary, assume that the centre of $D$ is at $a_0 + e^{x_0}$, for some $x_0 \isreal$. Define a branch of the logarithm, $L_1$, such that $\psi(w) = L_1(w - a_0)$ is a univalent map on $D$. Let $\phi$ be the univalent map $\phi = \psi \circ f$. Note that $\phi(\Gamma') = (-\infty, x_0)$, and (\[newfeq\]) holds with $r=1$. A contradiction follows by Lemma \[lem1\], since we have assumed that $|a_0|>R$. This establishes our second claim.

Finally we show that the projections of [[direct non-logarithmic singularities]{}]{} are bounded. This follows immediately from the fact that the projections of other types of [[transcendental singularities]{}]{} are bounded and from Theorem \[theo2\]. This completes the proof.

*Remark: It seems possible to generalise the result of Theorem \[theo1\] to transcendental meromorphic functions with direct tracts (see, for example, [@MR2439666] for more background on this concept). We have not done this here, for reasons of simplicity. However, the proof seems to work similarly, although a number of results used in this paper need to be generalised. In addition, we need to replace Theorem \[Heins\] with [@MR1703869 Corollary 1], and [@MR1196102 Lemma 1] with [@MR2439666 Lemma 6.3].\
*Acknowledgment: The author is grateful to Phil Rippon and Gwyneth Stallard for all their help with this paper.**
