A Schwarz lemma for locally univalent meromorphic functions

TL;DR

This paper establishes a sharp Schwarz lemma for meromorphic functions with bounded spherical derivative, leading to improved normality criteria, a dual extremal problem, and a generalization of Beurling's theorem in the spherical metric.

Contribution

It introduces a novel Schwarz-type lemma for meromorphic functions with bounded spherical derivative and connects it to extremal problems and classical theorems.

Findings

Proves a sharp Schwarz lemma for meromorphic functions with bounded spherical derivative.

Derives an improved normality criterion that is asymptotically optimal.

Generalizes Beurling's extension of the Riemann mapping theorem in the spherical metric.

Abstract

We prove a sharp Schwarz-type lemma for meromorphic functions with spherical derivative uniformly bounded away from zero. As a consequence we deduce an improved quantitative version of a recent normality criterion due to Grahl & Nevo and Steinmetz, which is asymptotically best possibe. Based on a well--known symmetry result of Gidas, Ni & Nirenberg for nonlinear elliptic PDEs, we relate our Schwarz-type lemma to an associated nonlinear dual boundary extremal problem. As an application we obtain a generalization of Beurling's extension of the Riemann mapping theorem for the case of the spherical metric.

Equations

$$f^{♯}(z):=\frac{|f^{\prime }(z)|}{1+|f(z){|}^2}$$

$${\mathcal{F}}_c:=\left\{f\in \mathcal{M}(\mathbb{D}):f^{♯}(z)\ge c\text{ for all }z\in \mathbb{D}\right\},$$

$$f^{♯}(z)\le \frac{1}{c{\left(1-|z{|}^2\right)}^2},z\in \mathbb{D},f\in {\mathcal{F}}_c.$$

$$|\eta |=1\text{ in }(a),|\eta |=\frac{1+\sqrt{1-4c^2}}{2c}\text{ in }(b),|\eta |=\frac{1-\sqrt{1-4c^2}}{2c}\text{ in }(c).$$

$$\frac{1-\sqrt{1-4c^2{\left(1-|z_0{|}^2\right)}^2}}{2c{\left(1-|z_0{|}^2\right)}^2}\le f^{♯}(z_0)\le \frac{1+\sqrt{1-4c^2{\left(1-|z_0{|}^2\right)}^2}}{2c{\left(1-|z_0{|}^2\right)}^2},z_0\in \mathbb{D}.$$

$$\underset{|z|\to 1}{\mathrm{limsup}}{\left(1-|z{|}^2\right)}^2{f}^{♯}(z)\le \frac{1}{c},f\in {\mathcal{F}}_c.$$

$$\underset{n\in \mathbb{N}}{\inf }{\left(1-|z_n{|}^2\right)}^2{f}^{♯}(z_n)>0$$

$$f^{♯}(z_0)\le {\left(\frac{\sqrt{4+|z_0{|}^2}-|z_0|}{2}\right)}^2\frac{1}{c{\left(1-|z_0{|}^2\right)}^2},z_0\in \mathbb{D}.$$

$$\underset{z\to \xi }{\lim }\frac{|f^{\prime }(z)|}{1+|f(z){|}^2}=c\text{ for all }|\xi |=1.$$

$$S(z)=\frac{z+z_0}{1+\overline{z_0}z},$$

$$u(z):=\log f^{♯}(S(z))=\log \frac{|f^{\prime }(S(z))|}{1+|f(S(z)){|}^2}$$

$$v_r(z):=\log \frac{|f^{\prime }(S(z))|}{1+{\left|\frac{f(S(z))}{z}r\right|}^2},0<r\le 1.$$

$$\Delta \log f^{♯}=-4f^{♯}(z{)}^2\text{ in }\mathbb{D},$$

$$v_r(z)=\log g_r^{♯}(z)+\log \left|\frac{f^{\prime }(S(z))}{g_r^{\prime }(z)}\right|,$$

$$g_r(z):=\frac{(f\circ S)(z)}{z}r\in \mathcal{M}(\mathbb{D}),$$

$$\Delta v_r(z)=-4g_r^{♯}(z{)}^2,z\in \mathbb{D}.$$

$$u=v_r\text{ on }|z|=r,$$

$$\underset{|z|\le r}{\inf }u(z)=\underset{|z|\le r}{\inf }{v}_r(z)\le v_r(0).$$

$$\log c\le \underset{z\in \mathbb{D}}{\inf }\log f^{♯}(z)=\underset{z\in \mathbb{D}}{\inf }u(z)=\underset{z\in \mathbb{D}}{\inf }{v}_1(z)\le v_1(0),$$

$$c\le \underset{z\in \mathbb{D}}{\inf }{f}^{♯}(z)\le \frac{|f^{\prime }(z_0)|}{1+{\left(1-|z_0{|}^2\right)}^2|f^{\prime }(z_0){|}^2}.$$

$$c{\left(1-|z_0{|}^2\right)}^2|f^{\prime }(z_0){|}^2-|f^{\prime }(z_0)|+c\le 0.$$

$$\varrho (x):=c(1-|z_0{|}^2{)}^2{x}^2-x+c$$

$$\frac{1\pm \sqrt{1-4c^2{\left(1-|z_0{|}^2\right)}^2}}{2c{\left(1-|z_0{|}^2\right)}^2},$$

$$c\le \frac{1}{2(1-|z_0{|}^2)}.$$

$$\frac{1-\sqrt{1-4c^2{\left(1-|z_0{|}^2\right)}^2}}{2c{\left(1-|z_0{|}^2\right)}^2}\le |f^{\prime }(z_0)|\le \frac{1+\sqrt{1-4c^2{\left(1-|z_0{|}^2\right)}^2}}{2c{\left(1-|z_0{|}^2\right)}^2}.$$

$$v_1(z)=\log \frac{|(f\circ S{)}^{\prime }(z)|}{1+{\left|(f\circ S)(z)/z\right|}^2}-\log |S^{\prime }(z)|$$

$$|\eta |=\frac{1\pm \sqrt{1-4c^2}}{2c}.$$

$$|w^{\prime }(z_0)|\le \frac{\sqrt{4+|z_0{|}^2}-|z_0|}{2\left(1-|z_0{|}^2\right)}.$$

$$w(z)=\frac{z-z_0}{1-\overline{z_0}z}g(z)$$

$$g^{\prime }(z_0)=-\frac{\overline{z_0}}{1-|z_0{|}^2}g(z_0).$$

Full text

**A Schwarz lemma for locally univalent

meromorphic functions

**

Richard Fournier, Daniela Kraus and Oliver Roth

In Memory of Stephan Ruscheweyh

††footnotetext: Mathematics Subject Classification (2000)  Primary 30C55

1 Introduction and main results

Let $\mathop{\mathcal{M}({\mathbb{D}})}$ denote the set of all meromorphic functions in the unit disk ${\mathbb{D}}$ of the complex plane ${\mathbb{C}}$. Marty’s fundamental normality criterion [12], see also [14, §3.3], says that a family $\mathcal{F}\subseteq\mathop{\mathcal{M}({\mathbb{D}})}$ is normal if and only if the family of spherical derivatives

[TABLE]

of all $f\in\mathcal{F}$ is locally bounded (above) in ${\mathbb{D}}$. Some years ago, J. Grahl and S. Nevo [9] proved the surprising result that for any $c>0$ the family

[TABLE]

consisting of all meromorphic functions in ${\mathbb{D}}$ with spherical derivative uniformly bounded from below, is also normal.

The original proof in [9] is fairly involved and is based on a sophisticated application of Zalcman’s lemma [16]. Combining the result of Grahl and Nevo with Marty’s criterion shows that a uniform lower bound for $f^{\sharp}$ has to result in a locally uniform upper bound for $f^{\sharp}$. In fact, such an upper bound has been given by Steinmetz [15], who proved that

[TABLE]

The approach in [15] is based on the theory of complex differential equations and leads in particular to a short proof of the result of Grahl and Nevo.

The main purpose of this paper is to prove the following Schwarz–type lemma for functions in the classes $\mathcal{F}_{c}$, which provides in particular a sharp form of inequality (1.1) for the point $z=0$ including a precise description of the extremal functions.

Theorem 1.1** **(Schwarz lemma for $\mathcal{F}_{c}$)

Let $c>0$ and $f\in\mathcal{F}_{c}$. Then the following hold.

• (a)

$c\leq 1/2$.

• (b)

$f^{\sharp}(0)\leq\displaystyle\frac{1+\sqrt{1-4c^{2}}}{2c}$.

• (c)

$f^{\sharp}(0)\geq\displaystyle\frac{1-\sqrt{1-4c^{2}}}{2c}$.

Equality holds in either case if and only if $f(z)=T(\eta z)$ where $T$ is a rigid motion of the Riemann sphere and $\eta\in{\mathbb{C}}$ such that

[TABLE]

In particular, $\mathcal{F}_{1/2}$ is precisely the set of rigid motions of the Riemann sphere.

The proof of Theorem 1.1 is deceptively simple and only uses the minimum principle for superharmonic functions. In addition, Theorem 1.1 immediately yields not only the normality criterion of Grahl and Nevo, but our method of proof also leads to a sharpening of the quantitative upper bound (1.1) for $f^{\sharp}$ as well as a corresponding lower bound:

Theorem 1.2

Suppose that $f\in\mathcal{F}_{c}$ for some $c>0$. Then

[TABLE]

Both estimates are sharp if and only if $z_{0}=0$.

Even though the right–hand inequality in (1.2) clearly improves (1.1), both inequalities lead to the same asymptotic estimate, namely

[TABLE]

Recent work of Gröhn [10, Theorem 3] shows that there is a function $f\in\bigcup_{c>0}\mathcal{F}_{c}$ such that

[TABLE]

for some sequence $(z_{n})$ in ${\mathbb{D}}$ with $|z_{n}|\to 1$. Hence, for sufficiently small values of $c>0$ inequality (1.3) is sharp up to a multiplicative constant. The following result provides an improved upper bound for this constant. In fact it shows that for all possible values of $c$ one can replace the number $1$ on the right–hand side of (1.3) by $(3-\sqrt{5})/2\approx 0.38$:

Theorem 1.3

Let $c\in(0,1/2)$. Then for any $f\in\mathcal{F}_{c}$,

[TABLE]

The proof of Theorem 1.3 is different from the proof of Theorem 1.2 and is based on complex differential equations and a refinement of the Schwarz–Pick lemma for bounded holomorphic functions (Lemma 2.1).

We return to the Schwarz lemma for $\mathcal{F}_{c}$ (Theorem 1.1) and briefly indicate the second main goal of this note: We shall show that the “interior” extremal problem solved by Theorem 1.1 is related to a “dual” nonlinear boundary value problem and thereby to Beurling’s extension [4] of the Riemann mapping theorem, see Section 3 below. In particular, we establish the following rigidity property of locally conformal maps with “length–preserving boundary distortion”:

Theorem 1.4

Let $c>0$ and $f\in\mathcal{M}({\mathbb{D}})$ be locally univalent such that $f(0)=0$ and

[TABLE]

Then $f(z)=\eta z$ for some $\eta\in{\mathbb{C}}$.

This answers the question raised by Kühnau in [11] (Remark after Satz 2’) who proved Theorem 1.4 under the additional condition that $f$ is holomorphic on a neighborhood of $\overline{{\mathbb{D}}}$ and $f({\mathbb{D}})\subseteq{\mathbb{D}}$, and asked if this condition is necessary. Now geometrically, it is natural to think of a function $f\in\mathcal{M}({\mathbb{D}})$ as a map $f:({\mathbb{D}},d_{{\mathbb{D}}})\to(\hat{{\mathbb{C}}},d_{\hat{{\mathbb{C}}}})$ from the unit disk ${\mathbb{D}}$ equipped with the hyperbolic distance $d_{{\mathbb{D}}}$ into the Riemann sphere $\hat{{\mathbb{C}}}$ equipped with the spherical distance $d_{\hat{{\mathbb{C}}}}$. We call such a map $f:({\mathbb{D}},d_{{\mathbb{D}}})\to(\hat{{\mathbb{C}}},d_{\hat{{\mathbb{C}}}})$ length-preserving on the circle $|z|=\varrho<1$ if for each subarc $\gamma$ of $|z|=\varrho$ the spherical length of $f(\gamma)$ is exactly the hyperbolic length of $\gamma$. Theorem 1.4 then has the following, perhaps appealing geometric interpretation, compare [11] for the special case $f({\mathbb{D}})\subseteq{\mathbb{D}}$.

Corollary 1.5

Let $f\in\mathcal{M}({\mathbb{D}})$ be a locally univalent function such that $f(0)=0$ and $0<\varrho<1$. Suppose that $f:({\mathbb{D}},d_{{\mathbb{D}}})\to(\hat{{\mathbb{C}}},d_{\hat{{\mathbb{C}}}})$ is length-preserving on the circle $|z|=\varrho$. Then $f(z)=\eta z$ for some $\eta\in{\mathbb{C}}$.

In fact, Theorem 1.1 shows that only $c\leq 1/2$ is possible in Theorem 1.4. A short computation (compare [11]) implies that this means that only $\varrho\leq\sqrt{2}-1$ is possible in Corollary 1.5. Hence there are locally univalent meromorphic maps $f:({\mathbb{D}},d_{{\mathbb{D}}})\to(\hat{{\mathbb{C}}},d_{\hat{{\mathbb{C}}}})$, $f(0)=0$, which are length–preserving on a circle $|z|=\varrho$ if and only if $\varrho\leq\sqrt{2}-1$. In contrast, there are “hyperbolically” length–preserving locally univalent maps $f:({\mathbb{D}},d_{{\mathbb{D}}})\to({\mathbb{D}},d_{{\mathbb{D}}})$ (analogously defined) of the form $f(z)=\eta z$ on any circle $|z|=\varrho\in(0,1)$.

The plan of this note is as follows. In Section 2 we prove Theorems 1.1, 1.2 and 1.3. In Section 3 we discuss the relation of the Schwarz lemma for $\mathcal{F}_{c}$ with Beurling’s extension of the Riemann mapping theorem and in particular prove Theorem 1.4. The main additional tool is a celebrated result of Gidas, Ni and Nirenberg [8] concerning positive solutions of certain semilinear elliptic PDEs. The paper concludes with Section 4, which presents a simple direct proof of the Grahl–Nevo normality criterion and a quantitative normality result for rational functions in $\mathcal{F}_{c}$ based on a Bernstein–type inequality for rational functions due to Borwein and Erdelyi [5].

2 Proof of Theorems 1.1, 1.2 and 1.3

Proof of Theorems 1.1 and 1.2.

Let $z_{0}\in{\mathbb{D}}$. Since postcomposing $f\in\mathcal{F}_{c}$ with a rigid motion of the Riemann sphere does not change the spherical derivative of $f$, we may assume that $f(z_{0})=0$.

(i)   We consider the unit disk automorphism

[TABLE]

and the auxiliary functions

[TABLE]

and

[TABLE]

Since $f$ is a locally univalent meromorphic function, the function $\log f^{\sharp}$ is smooth and in fact satisfies Liouville’s equation

[TABLE]

as it is well–known and also easy to show by direct computation. Hence, $u$ is superharmonic on ${\mathbb{D}}$. Since $S(0)=z_{0}$ and $f(z_{0})=0$, we see that also $v_{r}$ is a smooth function on ${\mathbb{D}}$. In view of

[TABLE]

where

[TABLE]

we see that

[TABLE]

Hence $v_{r}$ is a superharmonic function on ${\mathbb{D}}$ as well.

(ii)   Since for each $0<r<1$,

[TABLE]

the minimum principle applied to the superharmonic functions $u$ and $v_{r}$ shows

[TABLE]

Letting $r\to 1$, we get

[TABLE]

or, equivalently,

[TABLE]

Hence

[TABLE]

Now, the quadratic function

[TABLE]

has the zeros

[TABLE]

so it takes on nonpositive values if and only if

[TABLE]

In particular, this shows that $c\leq 1/2$ and

[TABLE]

If equality holds on either side of (2.3), then equality holds in both inequalities of (2.2). Therefore the minimum principle shows that $v_{1}$ is constant. In view of (2.1), we get that $g_{1}$ is constant and hence $f(S(z))=\eta z$ for some $\eta\in{\mathbb{C}}$. Now, using again that

[TABLE]

is constant, we see that $S^{\prime}$ is constant and therefore $z_{0}=0$. Hence (2.3) is sharp only for $z_{0}=0$. Now let $z_{0}=0$. Then equality holds on either side of (2.3) if and only if $f(z)=\eta z$ and

[TABLE]

If, in addition, $c=1/2$, then equality holds in (2.3) on both sides, so $|f^{\prime}(0)|=1$, that is, every $f\in\mathcal{F}_{c}$ such that $f(0)=0$ has the form $f(z)=\eta z$ with $|\eta|=1$. This completes the proof. ∎

The proof of Theorem 1.3 is based on the following simple Schwarz–Pick type lemma.

Lemma 2.1

Suppose that $z_{0}\in{\mathbb{D}}\setminus\{0\}$ and $w:{\mathbb{D}}\to{\mathbb{D}}$ is a holomorphic function such that $w(z_{0})=0$ and $w^{\prime\prime}(z_{0})=0$. Then

[TABLE]

Equality can hold only if $w$ is a Blaschke product of degree $2$.

In the proof we will identify all the extremal functions semi–explicitly. We intentionally have excluded the case $z_{0}=0$ in Lemma 2.1.

Proof.

Write

[TABLE]

for some holomorphic function $g:{\mathbb{D}}\to\overline{{\mathbb{D}}}$ and note that $g({\mathbb{D}})\subseteq{\mathbb{D}}$. Then $w^{\prime\prime}(z_{0})=0$ is equivalent to

[TABLE]

The Schwarz–Pick lemma applied to $g$ implies

[TABLE]

which is equivalent to

[TABLE]

Again by the Schwarz–Pick lemma, we see that equality occurs if and only if $g$ is a unit disk automorphism such that (2.4) holds. ∎

Proof of Theorem 1.3.

In view of Theorem 1.1 we may assume that $z_{0}\not=0$. We first closely follow the proof of (1.1) in [15] and assume $f(z_{0})=0$. Since $f\in\mathcal{M}({\mathbb{D}})$ is locally univalent its Schwarzian derivative

[TABLE]

is holomorphic in ${\mathbb{D}}$ and we can write

[TABLE]

with holomorphic functions $w_{1},w_{2}:{\mathbb{D}}\to{\mathbb{C}}$ both of which are solutions of the linear second order ODE

[TABLE]

Note that the Wronskian $w_{1}^{\prime}w_{2}-w_{1}w_{2}^{\prime}$ of $w_{1}$ and $w_{2}$ is constant, so by renormalizing the solutions we may assume that

[TABLE]

In particular, we get

[TABLE]

Since $f^{\sharp}(z)\geq c$, the first identity shows that $w:=\sqrt{c}\,w_{1}$ is a holomorphic selfmap of the unit disk with $w(z_{0})=0$ and

[TABLE]

since $S_{f}$ is holomorphic. Hence we are in a position to apply Lemma 2.1 and obtain

[TABLE]

∎

Remark 2.2

Using the standard Schwarz–Pick lemma

[TABLE]

instead of the “improved” Schwarz–Pick type Lemma 2.1 in the last step of the preceding proof gives the less precise inequality (1.1). This is exactly the proof of (1.1) given in [15]. It does not fully use the fact that $S_{f}$ is holomorphic.

3 The Schwarz lemma for $\mathcal{F}_{c}$ and nonlinear boundary value problems

In this section we show that the interior extremal problem solved in Theorem 1.1 can be related to a “dual” nonlinear boundary extremal problem. This establishes a link between the Schwarz lemma for $\mathcal{F}_{c}$ and a class of boundary value problems arising in conformal geometry which have first been investigated by Beurling [4].

The point of departure is the following peculiar property of the extremal functions in Theorem 1.1:

Theorem 3.1

Let $c>0$ and $f\in\mathop{\mathcal{M}({\mathbb{D}})}$ locally univalent. Then the following are equivalent:

• (a)

$\displaystyle\lim\limits_{|z|\to 1}f^{\sharp}(z)=c$.

• (b)

$c\leq 1/2$* and $f(z)=T(\eta z)$ with a rigid motion $T$ of the Riemann sphere and*

[TABLE]

In particular, this proves Theorem 1.4 which is merely a special case of Theorem 3.1.

Proof.

(b) $\Longrightarrow$ (a): This is just a computation.

(a) $\Longrightarrow$ (b): This is a simple application of a rather deep result of Gidas, Ni and Nirenberg [8], which has become a standard tool in elliptic PDE, in combination with a nonlinear version of the Schwarz reflection principle, see [13]. Let $f\in\mathop{\mathcal{M}({\mathbb{D}})}$ be locally univalent and satisfy condition (a). By [13, Theorem 1.8], we infer that $f$ has a meromorphic continuation to an open neighborhood of the closed unit disk $\overline{{\mathbb{D}}}$. This shows that

[TABLE]

is a $C^{2}$–function on $\overline{{\mathbb{D}}}$, that is, on an open neighborhood of $\overline{{\mathbb{D}}}$, such that

[TABLE]

By the minimum principle, the superharmonic function $u$ is positive on ${\mathbb{D}}$. Hence Theorem 1 in [8] forces $u$ to be radially symmetric,

[TABLE]

for some strictly decreasing $C^{2}$–function $v:[0,1]\to[0,\infty)$. It is now a simple matter to see that all radially symmetric solutions of the boundary value problem (3.2) have the form

[TABLE]

with $\eta\in{\mathbb{C}}$ as in (3.1). For convenience, we indicate the main steps. Since

[TABLE]

where $r=|z|$ and ′ indicates differentiation with respect to $r$, the strictly decreasing function $v\in C^{2}([0,1])$ solves

[TABLE]

We substitute $r=e^{x}$ and obtain for $w(x):=v(e^{x})+x+\log(2c)$ the initial value problem

[TABLE]

This ODE has $2\,w^{\prime}(x)$ as an integrating factor, so

[TABLE]

Integrating from $x=a$ to $x=t$ and using that

[TABLE]

as well as

[TABLE]

we arrive at

[TABLE]

In particular, $w(x)\leq 0$ for all $x\in(-\infty,0]$. This implies $\log(2c)=w(0)\leq 0$, and hence $c\leq 1/2$. Note also that there exists no subinterval $[a,b]\subseteq(-\infty,0]$ such that $w(a)=w(b)=0$ and $w(x)\not=0$ for all $x\in(a,b)$, since otherwise we would have $w^{\prime}(c)=0$ for some $c\in(a,b)$ and so $w(c)=0$ by (3.4). Since $w^{\prime\prime}=-e^{2w}$, the $C^{2}$–function $w$ cannot vanish identically on any open interval, so it follows that $w$ vanishes at one point at most. Therefore,

[TABLE]

with at most one change of sign on $(-\infty,0]$. The resulting two ODEs (one for each sign) are both separable. Hence if one is patient enough, elementary integration on each interval where $w\not=0$, would ultimately show that the solutions $u(z)$ to (3.2) have the form (3.3). However, one can avoid this lengthy calculation as follows. Let us first consider the case $c<1/2$. If $w(x)$ has no zero in $(-\infty,0]$, then $w<0$ there and in view of $w^{\prime}(x)\to 1$ as $x\to-\infty$ we see that

[TABLE]

Hence $w$ is uniquely determined by the standard uniqueness result for ODEs. If $w$ has exactly one zero $\alpha\in(-\infty,0)$, then $w$ is a solution of the initial value problem

[TABLE]

on $[\alpha,0]$, which has a unique solution on $(\alpha,0]$. Hence $\alpha$ and $w:[\alpha,0]\to{\mathbb{R}}$ are uniquely determined by $c$. Since $w(\alpha)=0$, we have $w^{\prime}(\alpha)=0$ by $(\ref{eq:ODE})$. Thus $w$ solves the initial value problem $w^{\prime\prime}=-e^{2w}$, $w(\alpha)=w^{\prime}(\alpha)=0$, and is therefore uniquely determined also on $(-\infty,\alpha]$. In the remaining case $w(0)=0$ and $c=1/2$, we have $w^{\prime}(0)=0$ and $w^{\prime\prime}=-e^{2w}$, so $w$ is uniquely determined as before. We therefore see that there are at most two solutions to (3.2) which coincide for $c=1/2$. This shows that there are no solutions $u(z)$ to (3.2) different from those given by (3.3) with $\eta\in{\mathbb{C}}$ as in (3.1).

∎

Theorem 3.2** **(The Schwarz lemma for $\mathcal{F}_{c}$ and a dual boundary extremal problem)

Let $c>0$ and $F\in\mathcal{F}_{c}$. Then the following are equivalent.

• (a)

$F$* is extremal for one of the interior extremal problems*

[TABLE]

• (b)

$F$* is extremal for the boundary extremal problem*

[TABLE]

The proof of Theorem 3.2 is by now obvious because we have identified all functions $F\in\mathcal{F}_{c}$ with property (a) in Theorem 1.1 and those with property (b) in Theorem 3.1 in an explicit way. It would be desirable to have a direct proof of the fact that (a) and (b) are equivalent.

Problem 3.3

Theorem 3.2 roughly says that every $f\in\mathcal{F}_{c}$ that maximizes/minimizes $f^{\sharp}$ at the origin actually minimizes $f^{\sharp}$ on the entire unit circle. Now suppose that $f\in\mathcal{F}_{c}$ maximizes/minimizes $f^{\sharp}$ over the set $\mathcal{F}_{c}$ at a point $z_{0}\not=0$. Does $f^{\sharp}$ have a corresponding boundary extremal property on (part of) the unit circle ?

We are now in a position to relate the Schwarz lemma for the class $\mathcal{F}_{c}$ with Beurling’s well–known extension of the Riemann mapping theorem (see [1, 2, 4, 6, 3, 7]). Denote by $\mathcal{H}_{0}({\mathbb{D}})$ the set of all holomorphic functions $g:{\mathbb{D}}\to{\mathbb{C}}$ with $g(0)=0$ and $g^{\prime}(0)>0$. For a given positive, continuous and bounded function $\Phi:{\mathbb{C}}\to{\mathbb{R}}$, Beurling [4] considered the nonlinear boundary value problem††This is a “Riemann–Hilbert–Poincaré problem”.

[TABLE]

and showed that this problem always admits univalent solutions $g\in\mathcal{H}_{0}({\mathbb{D}})$. In fact, Beurling even showed that there is always a kind of “maximal” resp. “minimal” univalent solution. In order to find the “minimal” univalent solution, Beurling considered the set of univalent “supersolutions” of (3.5),

[TABLE]

and proved in a first step that there is a unique function $g^{*}\in B_{\Phi}$ such that

[TABLE]

In a second step, he then showed that this “minimal” supersolution is in fact a solution of the boundary value problem (3.5). It appears that for Beurling’s method the assumption that $\Phi$ is bounded (or at least of sublinear growth as in [1]) is fairly essential. Now, it is easy to see that Beurling’s set of supersolutions for the unbounded function

[TABLE]

can be written as

[TABLE]

Since $\log g^{\sharp}$ is superharmonic for every (locally) univalent function $g$, we hence see that

[TABLE]

Therefore, Theorem 1.1 implies that for any $c\in(0,1/2]$ the function

[TABLE]

which belongs to $B_{\Phi_{c}}$, is the unique extremal function for the extremal problem

[TABLE]

and $g_{c}$ is obviously a solution to Beurling’s boundary value problem (3.5) for $\Phi=\Phi_{c}$. Clearly, an analogous result holds for the unique function in $B_{\Phi_{c}}$ which maximizes $g^{\prime}(0)$ for all $g\in B_{\Phi_{c}}$. By Theorem 3.1 and the superharmonicity of $\log g^{\sharp}$ for any locally univalent function $g\in\mathcal{H}_{0}({\mathbb{D}})$, these two solutions are the only two locally univalent solutions to (3.5) for $\Phi=\Phi_{c}$ ! To put it differently, Theorem 1.1 and Theorem 3.1 provide an extension of Beurling’s results at least for the specific function

[TABLE]

which is of quadratic and not merely sublinear growth.

For convenience, we state these considerations as a theorem, which as we have seen is now merely a restatement of Theorem 1.1 and Theorem 3.1.

Theorem 3.4** **(The Beurling–Riemann mapping theorem for the spherical

metric)

Suppose that $c>0$ and consider the boundary value problem

[TABLE]

for $g\in\mathcal{H}_{0}({\mathbb{D}})$.

• (a)

If $c<1/2$, then (3.6) has exactly two locally univalent solutions $g_{\pm}\in\mathcal{H}_{0}({\mathbb{D}})$ given by

[TABLE]

These solutions are univalent and they are the unique extremal functions for the extremal problems

[TABLE]

• (b)

If $c=1/2$, then $g(z)=z$ is the only locally univalent solution $g\in\mathcal{H}_{0}({\mathbb{D}})$ of (3.6).

• (c)

If $c>1/2$, then (3.6) has no locally univalent solution in $\mathcal{H}_{0}({\mathbb{D}})$.

We note that parts of Theorem 3.4 have been proved earlier by different means, see e.g. [1, 2, 7, 11]. The essential new ingredient is the uniqueness statement in part (a), which ultimately comes from the Gidas–Ni–Nirenberg theorem. For this uniqueness result the local univalence assumption is necessary. In fact, Kühnau [11, Satz 4] has shown that there are always additional solutions $g\in\mathcal{H}_{0}({\mathbb{D}})$ for Beurling’s boundary value problem (3.6) which, however, are not locally univalent.

4 Concluding remarks

Remark 4.1

In all of our results, the restriction to locally univalent functions is essential. The reason is that $\log f^{\sharp}$ is superharmonic only for locally univalent functions $f\in\mathop{\mathcal{M}({\mathbb{D}})}$, so the minimum principle can be applied and shows that

[TABLE]

In fact, the larger class

[TABLE]

is not even a normal family in view of the following example.

Example 4.2

Let

[TABLE]

Clearly ${g_{n}}\in\mathop{\mathcal{M}({\mathbb{D}})}$ and a straightforward computation leads to

[TABLE]

and hence $g_{n}\in\mathcal{G}_{c}$ for any $c\in(0,1/2)$ for all but finitely many $n$. However it is readily checked that $g_{n}(0)=0$ but

[TABLE]

on the punctured unit disk ${\mathbb{D}}\setminus\{0\}$. Hence none of the families $\mathcal{G}_{c}$, $c\in(0,1/2)$, is a normal family.

Remark 4.3

There is another simple proof of the Grahl–Nevo normality criterion. Let us set

[TABLE]

It is clear that for any $f\in\mathcal{F}_{c}$ and any $z\in{\mathbb{D}},$

[TABLE]

and by the fundamental normality test [14, p. 74] the family consisting of derivatives of functions $f\in\mathcal{F}_{c}$ is a normal family; note that this also follows from the plain fact that the family

[TABLE]

contains only functions analytic in the unit disk and is also uniformly bounded above there by (4.1). Now since for any $f$ in $\mathcal{F}_{c,0}$ we have

[TABLE]

we obtain (see e.g. Lemma 8 in [9]) that $\mathcal{F}_{c,0}$ is a normal family. We define for each $f\in\mathcal{F}_{c}$,

[TABLE]

so $F$ belongs to $\mathcal{F}_{c,0}$. Now let $\{f_{n}\}$ be a sequence in $\mathcal{F}_{c}$. Then $F_{n}\in\mathcal{F}_{c,0}$ for each $n$ and therefore $\{F_{n}\}$ has a subsequence $\{F_{n_{k}}\}$ which converges uniformly on compact subsets of ${\mathbb{D}}$. If $\{f_{n_{k}}(0)\}$ does not converge to the point at infinity, then passing to a further subsequence if necessary, we may assume that $\{f_{n_{k}}(0)\}\subseteq{\mathbb{C}}$ and that $f_{n_{k}}(0)\to\alpha\in{\mathbb{C}}$. Hence the sequence $\{f_{n_{k}}\}$ is compactly convergent in ${\mathbb{D}}$ because

[TABLE]

A similar line of reasoning is available if $\{f_{n_{k}}(0)\}$ does converge to the point at infinity and we may therefore conclude that $\mathcal{F}_{c}$ is a normal family.

Remark 4.4

It is sometimes possible to give straightforward proofs of the normality of specific subclasses of $\mathcal{F}_{c}$. Let $\mathcal{P}_{n}$ denote the class of complex polynomials of degree at most $n$ and $\mathcal{R}_{n}$ the class of rational functions $f=p_{n}/q_{n}$ with $p_{n}\in\mathcal{P}_{n}$ and

[TABLE]

where the points ${z_{j}}$ are fixed once for all with $|z_{j}|>1$ . We set

[TABLE]

According to an estimate of Borwein and Erdelyi [5],

[TABLE]

with

[TABLE]

and clearly

[TABLE]

In particular, if $f\in\mathcal{F}_{c}\cap\mathcal{R}_{n}$ and if $|f(z_{0})|=||f||$ for some $|z_{0}|=1$, then

[TABLE]

It follows that $c/k_{n}\leq 1/2$ and

[TABLE]

The family $\mathcal{F}_{c}\cap\mathcal{R}_{n}$ is therefore uniformly bounded on the unit disk and in particular normal there.

The authors would like to express their gratitude to an anonymous referee for his or her careful reading of this paper and many valuable suggestions.