# Chordal Loewner chains and Teichmüller spaces on the half-plane

Huaying Wei  Department of Mathematics and Statistics, Jiangsu Normal
University Xuzhou 221116, PR China Department of Mathematics, School of
Education, Waseda University Shinjuku, Tokyo 169-8050, Japan<EMAIL_ADDRESS>and  Katsuhiko Matsuzaki  Department of Mathematics, School of Education,
Waseda University Shinjuku, Tokyo 169-8050, Japan<EMAIL_ADDRESS>
###### Abstract.

We consider a univalent analytic function $f$ on the half-plane satisfying the
condition that the supremum norm of its (pre-)Schwarzian derivative vanishes
on the boundary. Under certain extra assumptions on $f$, we show that there
exists a chordal Loewner chain initiated from $f$ until some finite time, and
this Loewner chain defines a quasiconformal extension of $f$ over the boundary
such that its complex dilatation is given explicitly in terms of the
(pre-)Schwarzian derivative in some neighborhood of the boundary. This can be
regarded as the half-plane version of the corresponding result developed on
the disk by Becker and also the generalization of the Ahlfors–Weill formula.
As an application of this quasiconformal extension, we complete the
characterization of an element of the VMO-Teichmüller space on the half-plane
using the vanishing Carleson measure condition induced by the (pre-)Schwarzian
derivative.

###### Key words and phrases:

Loewner chain, evolution family, univalent function, Schwarzian derivative,
quasiconformal extension, Teichmüller space, Carleson measure, VMOA

###### 2020 Mathematics Subject Classification:

Primary 30C62, 30C80, 37F30; Secondary 30H35, 30C35, 30D05

Research supported by Japan Society for the Promotion of Science (KAKENHI
18H01125 and 21F20027).

##  1\. Introduction

Disks and half-planes in the complex plane are conformally equivalent. Most of
the results in complex analysis have no difference between these cases, and
thus one may easily overlook the situations where the difference occurs when
boundary conditions are imposed on functions of these domains; one boundary is
compact and the other is non-compact. In active research areas of complex
analysis interacting with each other, we can find these phenomena in recent
studies. One is a Loewner chain and the other is a Teichmüller space. In this
paper, we focus on a Loewner chain based on a point on a non-compact boundary,
and consider its application to a Teichmüller space whose functions satisfy a
certain vanishing condition on the non-compact boundary.

The idea of the Loewner theory, first introduced by Loewner [29] in 1923 and
later developed by Kufarev [25] in 1943 and Pommerenke [31] in 1965, is to
embed a univalent function into a family of time-parametrized univalent
functions (nowadays known as a Loewner chain) satisfying a suitable
differential equation. Subsequently, the parametric representation method of
univalent functions has been widely applied and developed in the theory of
univalent functions as well as in other fields of mathematics. These include
its stochastic analogue (Schramm–Loewner Evolution) discovered by Schramm [34]
in 2000, which has attracted substantial attention in probability theory and
conformal field theory. The most remarkable result in classical complex
analysis is that the famous Bieberbach conjecture was solved by de Branges
[17] in 1985 with the use of the extension of Loewner’s original approach.

In this paper, our interest is mainly devoted to an application of the Loewner
differential equation of chordal type to quasiconformal extensions of
univalent functions on the right half-plane
$\mathbb{H}:=\\{z=x+iy\in\mathbb{C}:x>0\\}$. The chordal case was less well
known until a decade ago, while the radial case is the main focus of the
classical Loewner theory.

Pommerenke applied successfully the radial Loewner differential equation to
show the univalence criteria of analytic functions on the unit disk
$\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ (see [32, Theorem 6.2]).
Particularly, it is worth mentioning that many sufficient conditions for
univalence can be deduced in this way (see [10, Theorem 5.3]). Moreover, most
of the univalence criteria can be refined to quasiconformal extension
criteria.

We first recall the following well-known result prior to the application of
the Loewner theory. Let $f$ be analytic in a disk $D$ in the extended complex
plane $\overline{\mathbb{C}}$, and assume that

$\sup_{z\in D}\frac{1}{2}\rho_{D}^{-2}(z)|Sf(z)|\leq k.$ (1.1)

Here, $\rho_{D}$ denotes the hyperbolic density on $D$ (of the Gaussian
curvature $\equiv-4$), which satisfies
$\rho_{D}(\varphi(z))|\varphi^{\prime}(z)|=1/(2{\rm Re}z)$ for a conformal
mapping $\varphi$ of $\mathbb{H}$ onto $D$; $Sf=(Pf)^{\prime}-(Pf)^{2}/2$ is
the Schwarzian derivative of $f$ given by the pre-Schwarzian derivative
$Pf:=f^{\prime\prime}/f^{\prime}$. If $k=1$ then $f$ is univalent in $D$,
which is a sharp univalence criterion proved by Nehari [30] in 1949; if $k<1$
then $f$ admits a $k$-quasiconformal extension to $\overline{\mathbb{C}}$ as
Ahlfors and Weill [4] found in 1962 (see also [27, p.87]). In fact, when
$D=\mathbb{D}$, the complex dilatation $\mu(z)$ of this extension has the form

$\mu(1/\bar{z})=-\frac{1}{2}\left(z/\bar{z}\right)^{2}(1-|z|^{2})^{2}Sf(z);$
(1.2)

when $D=\mathbb{H}$, it has the form

$\mu(-\bar{z})=-\frac{1}{2}(2{\rm Re}z)^{2}Sf(z).$ (1.3)

These results can be also obtained by using the Loewner chain. In 1972, Becker
[8] discovered the method of showing the quasiconformal extendibility of
univalent functions in $\mathbb{D}$ by means of the radial Loewner
differential equation: let $(f_{t})_{t\geq 0}$ be a univalent solution of

$\dot{f}_{t}(z)=zp(z,t)f_{t}^{\prime}(z)$

for all $z\in\mathbb{D}$ and almost every $t\geq 0$ with some Herglotz
function $p(z,t)$. He showed that if the image domain $p(\mathbb{D},t)$ lies
in a closed disk

$U(k):=\left\\{w\in\mathbb{H}:\;\left|w-\frac{1+k^{2}}{1-k^{2}}\right|\leq\frac{2k}{1-k^{2}}\right\\}$

for almost every time $t\geq 0$ and for some $k<1$ independent of $t$, then
each function $f_{t}$ has a quasiconformal extension to the whole plane
$\mathbb{C}$. More recently, Gumenyuk and Hotta [22] developed this method in
the chordal case by considering the corresponding equation

$\dot{f}_{t}(z)=-p(z,t)f_{t}^{\prime}(z),$

and opened a way of quasiconformal extensions of analytic functions defined on
$\mathbb{H}$. The advantage of the Loewner theory is that these extensions are
yielded explicitly by tracing the boundary values of the Loewner chain.

There is another advantage of using the Loewner chain. As the existence of a
quasiconformal extension depends only on the boundary curve of the image
domain, it is natural to consider the limit at the boundary corresponding to
condition (1.1). In this case, one has to assume first that $f$ is already
known to be univalent having a Jordan domain as its image. Towards this
direction, we refer to Becker’s work (see [9, 10]). As we mentioned above, he
obtained a method for global quasiconformal extendibility. Moreover, this
implies that if the time-variable $t$ is taken only over an interval
$[0,\tau)$ then the extension only to the disk $\\{z\in\mathbb{C}:|z|<R\\}$
for some $R>1$ can be obtained. Based on that, the following sufficient
condition for quasiconformal extendibility of univalent analytic functions was
proved. See [9, Theorem 1], and also [10, Theorem 5.4] and [11, Theorem 3].

######  Theorem 1.1 (Becker).

Let $f$ be univalent and analytic in $\mathbb{D}$ such that $f(\mathbb{D})$ is
a Jordan domain. If

$\limsup_{|z|\to 1^{-}}\frac{1}{2}(1-|z|^{2})^{2}|Sf(z)|\leq k\;(<1),$ (1.4)

then $f$ admits a quasiconformal extension to the disk
$\\{z\in\mathbb{C}:|z|<R\\}$ for some $R>1$ whose complex dilatation $\mu$ has
the form in (1.2).

We note that under the same assumption of the above theorem, $f$ given on a
smaller disk $\\{z\in\mathbb{C}:|z|<R-\varepsilon\\}$ for any $\varepsilon>0$
can be further extended quasiconformally to $\mathbb{C}$ by a general fact of
quasiconformal extension as in [28, Theorem II.1.8].

In this paper, we will give, by means of the chordal Loewner differential
equation, the half-plane analogue of Theorem 1.1, which is also the boundary
limit version in the half-plane case of the Ahlfors–Weill extension:

######  Theorem 1.2.

Let $h$ be univalent and analytic in $\mathbb{H}$ with
$\lim_{z\to\infty}h(z)=\infty$ such that $h(\mathbb{H})$ is a Jordan domain.
Let $h$ satisfy the condition

$\lim_{{\rm Re}z\to 0^{+}}\frac{1}{2}(2{\rm Re}z)^{2}\left|Sh(z)\right|=0.$
(1.5)

Suppose further that either

  1. $\mathbf{(A)}$

both $h$ and $h^{-1}$ are locally uniformly continuous in a neighborhood of
$\infty$, or

  2. $\mathbf{(B)}$

$h$ is locally quasiconformally extendible to a neighborhood of $\infty$.

Then, the function

$\hat{h}(z)=\begin{cases}h(z),&\text{if}\;\;{\rm Re}\,z\geq 0,\\\
h(z^{*})+\frac{(2{\rm Re}z)h^{\prime}(z^{*})}{1-({\rm
Re}\,z)Ph(z^{*})}\quad(z^{*}:=-\bar{z}),&\text{if}\;\;-\tau<{\rm
Re}\,z<0,\end{cases}$ (1.6)

defines a quasiconformal extension of $h$ over i$\mathbb{R}$ to
$\mathbb{H}^{*}_{(0,\tau)}:=\\{z\in\mathbb{C}:-\tau<{\rm Re}\,z<0\\}$ for some
$\tau>0$ such that its complex dilatation $\mu$ on $\mathbb{H}^{*}_{(0,\tau)}$
has the form

$\mu(z)=-\frac{1}{2}(2{\rm Re}z)^{2}Sh(z^{*}).$ (1.7)

Condition $\mathbf{(A)}$ is a practical assumption. For instance, if $h$
satisfies the hydrodynamic normalization at $\infty$ uniformly in $\mathbb{H}$
(see [21]), then this condition follows. Condition $\mathbf{(B)}$ is an a
priori assumption in the sense that if $h$ admits such a quasiconformal
extension as in the conclusion of the theorem, then this condition must be
satisfied.

We remark that if assumptions $\mathbf{(A)}$ and $\mathbf{(B)}$ are not
imposed then the assertion of Theorem 1.2 does not hold true. This will be
verified by constructing an explicit example in Corollary 3.3. This, in
particular, illustrates that the case of half-plane shown in Theorem 1.2
represents a more complicated object where we have to deal with compactness
problems on the boundary that do not arise in the case of disk. In Section 3,
we develop the arguments for this example.

We demonstrate the proof of Theorem 1.2 in Section 4 by relying on the Loewner
theory. In Section 2, we collect several necessary definitions and results
from this theory which will be used in the proofs as well as a concise summary
of the Loewner theory from a modern viewpoint.

Under the circumstances of Theorem 1.2, a family of analytic functions
initiated from $h$ is defined canonically by

$h_{t}(z)=h(z+t)-\frac{2th^{\prime}(z+t)}{1+tPh(z+t)}$

for $z\in\mathbb{H}$ and for $t$ within a limited period of time starting at
$0$. An important step for the proof of Theorem 1.2 is to show that $h_{t}$ is
univalent when $t\geq 0$ is sufficiently small, which turns out to be a
Loewner chain over an interval $[0,\tau)$ for some $\tau>0$. Gumenyuk and
Hotta [22] carried out this under the circumstances where the time $t$ can be
extended to $+\infty$, but their argument does not work for the case of
limited time period. We remark that, differently from the case of half-plane,
the corresponding argument for Theorem 1.1 is essentially the same as that for
the case where the time $t$ extends to $+\infty$. Our method of showing that
$h_{t}$ is univalent in a small interval of time involves novel techniques.
This is achieved in Theorem 4.3 under assumption $\mathbf{(A)}$ and Theorem
4.5 under assumption $\mathbf{(B)}$.

For the proof, we need to show that condition (1.5) is equivalent to

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)\left|Ph(z)\right|=0.$ (1.8)

In Section 3, we prove this equivalence in Theorem 3.5 which seems to be
natural, but is missing in the literature. Moreover, by replacing condition
(1.5) with (1.8), we can also obtain the analogous result as Theorem 5.1 to
Theorem 1.2, which represents the complex dilatation $\mu$ in terms of $Ph$.
This is stated briefly in Section 5.

As a special case of Theorem 1.2 and also of Theorem 5.1, the following form
can be used conveniently for subspaces of the universal Teichmüller space
because univalent analytic functions are already assumed to be
quasiconformally extendible to $\overline{\mathbb{C}}$ in these settings. The
importance of this result lies not on the quasiconformal extendibility of
course, but on the representation of the complex dilatation.

######  Corollary 1.3.

Let $h$ be univalent and analytic in $\mathbb{H}$ that is quasiconformally
extendible to $\mathbb{C}$ with $\lim_{z\to\infty}h(z)=\infty$. If $h$
satisfies (1.5), then the assertion in Theorem 1.2 holds, that is, it has the
extension whose complex dilatation is of the form in (1.7) in a strip domain
$\mathbb{H}^{*}_{(0,t)}$ for some $t>0$ over $i\mathbb{R}$, and further
extendible to $\mathbb{C}$ keeping this initial extension.

Thanks to the fact that the complex dilatation $\mu$ of the quasiconformal
extension $\hat{h}$ in Theorem 1.2 is expressed explicitly by the Schwarzian
derivative $Sh$ of $h$, and the way of constructing $\hat{h}$ by means of the
boundary values of a suitable Loewner chain, the above Corollary 1.3 can be
neatly used to give an appropriate quasiconformal extension for the subspaces
of the universal Teichmüller space smaller than the so-called little universal
Teichmüller space on the half-plane. In this paper, we deal with the VMO-
Teichmüller space as such an example.

The universal Teichmüller space $T$ can be regarded as the set of all
conformal mappings $h$ (up to the composition of a Möbius transformation) on
$\mathbb{H}$ which can be extended to a quasiconformal homeomorphism of
$\overline{\mathbb{C}}$. Subspaces of $T$ are obtained by imposing some
conditions on $h$ in general. If this condition is conformally invariant, then
there is no difference between the cases where we consider it on $\mathbb{D}$
and on $\mathbb{H}$. The Weil–Petersson Teichüller space and the BMO-
Teichmüller space are such examples, which have received much attention over
the years (see [3, 13, 35, 36, 39, 40] and references therein).

On the contrary, if the condition is given on the boundary, for instance, the
vanishing condition of the supremum norm of Schwarzian derivative as for the
little universal Teichmüller space, Teichmüller spaces on $\mathbb{D}$ and on
$\mathbb{H}$ are different. Becker and Pommerenke [12] worked on this for
$\mathbb{D}$, and later Hu, Wu and Shen [24] considered this on $\mathbb{H}$.

The VMO-Teichmüller space is the subspace of the BMO-Teichmüller space defined
by certain vanishing condition on the boundary (see [36]). If this space is
defined on the half-plane $\mathbb{H}$, it is the set of all $h\in T$
satisfying $\log h^{\prime}\in{\rm VMOA}(\mathbb{H})$, the space of analytic
functions in $\mathbb{H}$ of vanishing mean oscillation. In Section 6, as an
application of Corollary 1.3, we will prove the following result. This fills
in the missing part left in the study of various models of VMO-Teichmüller
space on the half-plane (see [35, Theorem 2.2], [39]).

######  Theorem 1.4.

Let $h$ be univalent and analytic in $\mathbb{H}$ such that $h$ is
quasiconformally extendible to $\overline{\mathbb{C}}$ with
$h(\infty)=\infty$. If $\log h^{\prime}\in{\rm VMOA}(\mathbb{H})$, then $h$
admits a quasiconformal extension to $\overline{\mathbb{C}}$ such that its
complex dilatation $\tilde{\mu}$ induces a vanishing Carleson measure

$\lambda_{\tilde{\mu}}:=\frac{|\tilde{\mu}(z)|^{2}}{(-2{\rm Re}z)}dxdy$ (1.9)

of the left half-plane $\mathbb{H}^{*}:=\\{z=x+iy\in\mathbb{C}:x<0\\}$.

##  2\. Preliminaries on the Loewner theory

In this section, we give several basic definitions and results on the
(generalized) Loewner theory, proposed by Bracci, Contreras and Díaz-Madrigal
[6, 7], which allows us to treat evolution families with inner fixed points
(the radial case) and with boundary fixed points (the chordal case) at the
same time. This unified theory relies partially on the theory of one
parametric semigroups, which is actually the autonomous version of the Loewner
theory.

###  2.1. Generalized Loewner theory

Let $D\subset\overline{\mathbb{C}}$ be a simply connected domain conformally
equivalent to $\mathbb{D}$ and $\mathbb{H}$. We denote the family of all
analytic functions on $D$ by ${\rm Hol}(D,\mathbb{C})$, and the family of all
analytic self-maps of $D$ by ${\rm Hol}(D)$.

Let $\mathcal{U}\subset{\rm Hol}(\mathbb{D})$ be a semigroup with respect to
the operation of composition containing the identity map ${\rm
id}_{\mathbb{D}}$. A family $(\phi_{t})_{t\geq 0}$ in $\mathcal{U}$ such that
$\phi_{0}={\rm id}_{\mathbb{D}}$, $\phi_{t}\circ\phi_{s}=\phi_{t+s}$ for any
$s,t\geq 0$, and $\phi_{t}(z)\to z$ locally uniformly on $\mathbb{D}$ as $t\to
0$, is called a continuous one-parameter semigroup. It is known [14] that for
any such semigroup $(\phi_{t})$ there exists a function $G\in{\rm
Hol}(\mathbb{D},\mathbb{C})$ such that for each $z\in\mathbb{D}$ the function
$\phi_{t}(z)$ is the unique solution of the initial value problem

$\frac{dw(z,t)}{dt}=G(w(z,t)),\quad w(z,0)=z.$ (2.1)

The function $G$ is called the infinitesimal generator of $(\phi_{t})$. A
criterion for a function $G\in{\rm Hol}(\mathbb{D},\mathbb{C})$ to be an
infinitesimal generator of some continuous one-parameter semigroup is the
following Berkson–Porta representation:

$G(z)=(\tau-z)(1-\bar{\tau}z)\,p(z)$ (2.2)

for a point $\tau\in\overline{\mathbb{D}}$ and a function $p\in{\rm
Hol}(\mathbb{D},\mathbb{C})$ with ${\rm Re}\,p(z)\geq 0$ for all
$z\in\mathbb{D}$. Moreover, if $G\not\equiv 0$, then such a representation is
unique. In fact, the point $\tau$ is the common Denjoy–Wolff point (see below
for its definition) of all $\phi_{t}$’s that are different from ${\rm
id}_{\mathbb{D}}$. Such a correspondence between the continuous one-parameter
semigroup $(\phi_{t})$ and the infinitesimal generator $G$ is one-to-one.

Now we introduce the Herglotz vector field in $\mathbb{D}$, which can be
regarded as a time-dependent infinitesimal generator. One of its equivalent
definitions is as follows (see [7, Theorem 4.8]). A Herglotz vector field in
$\mathbb{D}$ is a map $G:\;\mathbb{D}\times[0,+\infty)\to\mathbb{C}$ of the
form

$G(z,t)=(\tau(t)-z)(1-\overline{\tau(t)}z)\,p(z,t)$ (2.3)

for all $z\in\mathbb{D}$ and almost every $t\in[0,+\infty)$, where
$\tau:[0,+\infty)\to\overline{\mathbb{D}}$ is a measurable function and
$p:\mathbb{D}\times[0,+\infty)\to\mathbb{C}$ is a Herglotz function defined as
follows.

######  Definition.

A Herglotz function on $\mathbb{D}$ is a map
$p:\mathbb{D}\times[0,+\infty)\to\mathbb{C}$ satisfying the following
conditions:

  1. (HF1)

$p(z,\cdot)$ is locally integrable on $[0,+\infty)$ for all $z\in\mathbb{D}$;

  2. (HF2)

$p(\cdot,t)$ is analytic on $\mathbb{D}$ for almost every $t\in[0,+\infty)$
and ${\rm Re}\,p(\cdot,t)\geq 0$.

Moreover, [7, Theorem 4.8] asserts that if two couples $(p_{1},\tau_{1})$ and
$(p_{2},\tau_{2})$ generate the same $G(z,t)$ up to a set of measure zero on
the $t$-axis, then $p_{1}(z,t)=p_{2}(z,t)$ for all $z\in\mathbb{D}$ and almost
every $t\in[0,+\infty)$, and $\tau_{1}(t)=\tau_{2}(t)$ for almost every
$t\in[0,+\infty)$ satisfying $G(\cdot,\,t)\not\equiv 0$.

Furthermore, it is known that the evolution family $(\varphi_{s,t})_{t\geq
s\geq 0}$ is the unique solution of the following initial value problem for
the generalized Loewner–Kufarev ODE that is driven by the Herglotz vector
field $G$ (see [16, Section 2.1], [26, Chapter 18]):

$\frac{d\varphi_{s,t}(z)}{dt}=G(\varphi_{s,t}(z),\,t),\quad\varphi_{s,s}(z)=z$
(2.4)

for any initial point $z\in\mathbb{D}$, any starting time $s\geq 0$ and almost
every $t\geq s$. This equation establishes a one-to-one correspondence between
the evolution families $(\varphi_{s,t})$ and the Herglotz vector fields $G$ up
to a set of measure zero on the $t$-axis (see [7, Theorem 1.1]). Comparing
(2.1) and (2.4), we can regard an evolution family as a non-autonomous
analogue of a continuous one-parameter semigroup.

An independent definition of an evolution family can be given as follows:

######  Definition ([7]).

An evolution family in $\mathbb{D}$ is a two-parameter family
$(\varphi_{s,t})_{t\geq s\geq 0}\subset{\rm Hol}(\mathbb{D})$ satisfying the
following conditions:

  1. (EF1)

$\varphi_{s,s}={\rm id}_{\mathbb{D}}$ for all $s\geq 0$;

  2. (EF2)

$\varphi_{s,t}=\varphi_{u,t}\circ\varphi_{s,u}$ whenever $0\leq s\leq u\leq
t<+\infty$;

  3. (EF3)

for each $z\in\mathbb{D}$ there exists a non-negative locally integrable
function $k_{z}$ on $[0,\,+\infty)$ such that
$|\varphi_{s,u}(z)-\varphi_{s,t}(z)|\leq\int_{u}^{t}k_{z}(\xi)d\xi$ whenever
$0\leq s\leq u\leq t<+\infty$.

For all $0\leq s\leq t<+\infty$, $\varphi_{s,t}$ is univalent in $\mathbb{D}$,
which follows from the uniqueness of the solution of (2.4) (see [7, Corollary
6.3]) even though the definition does not require elements of an evolution
family to be univalent. The notion of a Loewner chain in the same framework as
an evolution family can be given as follows.

######  Definition ([15]).

A family $(f_{t})_{t\geq 0}\subset{\rm Hol}(\mathbb{D},\,\mathbb{C})$ is
called a Loewner chain if it satisfies the following conditions:

  1. (LC1)

each function $f_{t}$ is univalent for all $t\geq 0$;

  2. (LC2)

$f_{s}(\mathbb{D})\subset f_{t}(\mathbb{D})$ whenever $0\leq s\leq t<+\infty$;

  3. (LC3)

for any compact subset $K\subset\mathbb{D}$ and any $T>0$ there exists a non-
negative integrable function $k_{K,T}$ on $[0,\,T]$ such that
$|f_{s}(z)-f_{t}(z)|\leq\int_{s}^{t}k_{K,T}(\xi)d\xi$ whenever $z\in K$ and
$0\leq s\leq t\leq T$.

######  Remark.

In [7] and [15], the definitions of evolution families and Loewner chains
contain an integrability order $d\in[1,+\infty]$. We only need to consider the
most general case of order $d=1$.

For a given Loewner chain $(f_{t})$, the equation
$\varphi_{s,t}:=f_{t}^{-1}\circ f_{s}$ defines an evolution family.
Differentiating both sides with respect to $t$ yields that
$f_{t}^{\prime}(\varphi_{s,t})\dot{\varphi}_{s,t}+\dot{f}_{t}(\varphi_{s,t})=0$
where $f_{t}^{\prime}(z):=df_{t}(z)/dz$, $\dot{f}_{t}:=df_{t}/dt$, and
$\dot{\varphi}_{s,t}:=d\varphi_{s,t}/dt$. Combined with (2.4), this yields the
generalized Loewner–Kufarev PDE:

$\dot{f}_{t}(z)=-f_{t}^{\prime}(z)G(z,t)$ (2.5)

for all $z\in\mathbb{D}$ and almost every $t\geq 0$. Conversely, given an
evolution family $(\varphi_{s,t})$, or equivalently, a Herglotz vector field
$G$, one can obtain the corresponding Loewner chain $(f_{t})$ by (2.5), which
turns out to be unique up to the post-composition of a conformal mapping of
the domain $\bigcup_{t\geq 0}f_{t}(\mathbb{D})$. Moreover, the so-called
standard Loewner chain defined in [15] is determined uniquely, which is a
Loewner chain $(f_{t})$ such that $f_{0}(0)=f_{0}^{\prime}(0)-1=0$ (notice
that only $f_{0}$ is normalized) and $\bigcup_{t\geq 0}f_{t}(\mathbb{D})$ is
either $\mathbb{C}$ or a disk centered at the origin.

Overall, we have seen the one-to-one correspondences among evolution families
$(\varphi_{s,t})$, Herglotz vector fields $G$, couples $(p,\tau)$ of Herglotz
functions $p$ and measurable functions $\tau$, and Loewner chains $(f_{t})$ up
to conformal mappings. In particular, the relationship between
$(\varphi_{s,t})$ and $(p,\tau)$ is expressed by

$\frac{d\varphi_{s,t}(z)}{dt}=(\tau(t)-\varphi_{s,t}(z))(1-\overline{\tau(t)}\varphi_{s,t}(z))p(\varphi_{s,t}(z),t),\quad\varphi_{s,s}(z)=z$
(2.6)

for all $z\in\mathbb{D}$ and almost every $t\geq s\geq 0$. Moreover, the
relationship between $(f_{t})$ and $(p,\tau)$ is expressed by

$\dot{f}_{t}(z)=-f_{t}^{\prime}(z)(\tau(t)-z)(1-\overline{\tau(t)}z)p(z,t)$
(2.7)

for all $z\in\mathbb{D}$ and almost every $t\geq 0$. We point out that if
$\tau(t)\equiv{\rm const}$, then such a constant is the Denjoy–Wolff point of
$\varphi_{s,t}$ for all $0\leq s\leq t<+\infty$ different from ${\rm
id}_{\mathbb{D}}$ (see [7, Corollary 7.2]). This is the same as in the
Berkson–Porta representation for continuous one-parameter semigroups.

We end this subsection with a brief introduction of the Denjoy–Wolff point
mentioned above. The Julia–Wolff–Carathéodory theorem (see [18, Theorem 1.10])
asserts that for any $\varphi\in{\rm Hol}(\mathbb{D})$ different from ${\rm
id}_{\mathbb{D}}$ there exists a unique fixed point
$\tau\in\overline{\mathbb{D}}$ such that $\varphi(\tau)=\tau$ and
$|\varphi^{\prime}(\tau)|\leq 1$. Such $\tau$ is called the Denjoy–Wolff point
of $\varphi$. In the case of $\tau\in\partial\mathbb{D}$, these should be
understood as an angular limit and an angular derivative denoted by using
$\angle\lim$. Namely, $\varphi(\tau)=\tau$ means
$\angle\lim_{z\to\tau}\varphi(z)=\tau$, and this condition in fact implies
that $\varphi^{\prime}(\tau):=\angle\lim_{z\to\tau}(\varphi(z)-\tau)/(z-\tau)$
exists in $(0,\,+\infty]$ (see [33, Proposition 4.13]). Thus, if
$\tau\in\partial\mathbb{D}$ is the Denjoy–Wolff point of $\varphi$ then
$0<\varphi^{\prime}(\tau)\leq 1$. In the case of $\tau\in\mathbb{D}$, the
assertion $|\varphi^{\prime}(\tau)|\leq 1$ follows from $\varphi(\tau)=\tau$
by the Schwarz–Pick lemma.

The Denjoy–Wolff theorem asserts that, excluding a trivial exception where
$\varphi$ is ${\rm id}_{\mathbb{D}}$ or an elliptic automorphism of
$\mathbb{D}$, the sequence $\\{\varphi_{n}\\}$ of iterates of $\varphi$,
defined as $\varphi_{n}:=\varphi\circ\varphi_{n-1}$, $\varphi_{1}=\varphi$,
converges to the Denjoy–Wolff point $\tau$ uniformly on compact subsets of
$\mathbb{D}$. This is a prototype of hyperbolic dynamics and contributed much
to modern studies on dynamical systems (see [1, Appendix G]).

###  2.2. (Classical) radial Loewner theory

Hereafter, we consider the case that $\tau(t)$ is a constant function, and
thus it is the common interior Denjoy–Wolff point $\tau$ of the evolution
family $(\varphi_{s,t})$ in $\mathbb{D}$. When $\tau\in\mathbb{D}$, we can
assume that $\tau$ is $0$ by the conjugation of an automorphism of
$\mathbb{D}$ sending $\tau$ to $0$.

######  Definition.

An evolution family $(\varphi_{s,t})$ in $\mathbb{D}$ is said to be of radial
type, if for all $0\leq s\leq t<+\infty$, all elements $\varphi_{s,t}$’s
different from ${\rm id}_{\mathbb{D}}$ share the common interior Denjoy–Wolff
point at $0$. A Loewner chain $(f_{t})$ in $\mathbb{D}$ is said to be of
radial type if the corresponding evolution family
$(\varphi_{s,t}):=(f_{t}^{-1}\circ f_{s})$ is of radial type.

In this case, we see by the equation $\varphi_{s,t}=f_{t}^{-1}\circ f_{s}$
that a radial Loewner chain $(f_{t})$ is actually a Loewner chain such that
$f_{t}(0)=f_{0}(0)$ for all $t\geq 0$, and moreover, the Herglotz vector field
is $G(z,t)=-zp(z,t)$ for all $z\in\mathbb{D}$ and almost all $t\geq 0$. In
view of this, any radial evolution family $(\varphi_{s,t})$ and the
corresponding Loewner chain $(f_{t})$ satisfy the following radial
Loewner–Kufarev ODE and PDE, respectively:

$\frac{d\varphi_{s,t}(z)}{dt}=-\varphi_{s,t}(z)p(\varphi_{s,t}(z),t),\quad\varphi_{s,s}(z)=z$
(2.8)

for all $z\in\mathbb{D}$ and almost every $t\geq s\geq 0$, and

$\dot{f}_{t}(z)=zp(z,t)f_{t}^{\prime}(z)$ (2.9)

for all $z\in\mathbb{D}$ and almost every $t\geq 0$ with some Herglotz
function $p$ determined uniquely up to a set of measure zero on the $t$-axis.

In modern literature, the so-called classical radial evolution families and
Loewner chains mean a special kind of evolution families and Loewner chains
considered by Pommerenke (see [33, Chapter 6]).

######  Definition.

A radial evolution family and a radial Loewner chain are said to be classical
if they satisfy extra normalization hypotheses
$\varphi_{s,t}^{\prime}(0)=e^{s-t}$ and $f_{t}(0)=0$,
$f_{t}^{\prime}(0)=e^{t}$ for all $0\leq s\leq t<+\infty$, respectively.

Accordingly, the Herglotz function $p$ satisfies the extra condition
$p(0,t)=1$ for almost every $t\geq 0$. Moreover, the condition
$f_{t}^{\prime}(0)=e^{t}$ for all $t\geq 0$ ensures $\bigcup_{t\geq
0}f_{t}(\mathbb{D})=\mathbb{C}$, which gives the one-to-one correspondence
between the classical radial evolution family $(\varphi_{s,t})$ and the
classical radial Loewner chain $(f_{t})$ that is standard. Moreover, $(f_{t})$
can be recovered from $(\varphi_{s,t})$ by the equation
$f_{s}=\lim_{t\to+\infty}e^{t}\varphi_{s,t}$.

###  2.3. Chordal Loewner theory

Consider an evolution family $(\varphi_{s,t})$ in $\mathbb{D}$ such that for
all $0\leq s\leq t$, all elements $\varphi_{s,t}$’s different from ${\rm
id}_{\mathbb{D}}$ share the common boundary Denjoy–Wolff point at $\tau$. As
before, we can assume that $\tau$ is $1$ by the conjugation of a rotation of
$\mathbb{D}$. Obviously, the associated Herglotz vector field $G(z,t)$ has the
form $(1-z)^{2}p(z,t)$ for all $z\in\mathbb{D}$ and almost all $t\geq 0$ with
some Herglotz function $p$.

In the literature, the case of boundary fixed points is usually treated in the
half-plane model instead of the unit disk model because there the associated
vector field assumes a simpler form (see (2.10) below). Passing from the unit
disk $\mathbb{D}$ to the right half-plane $\mathbb{H}$ by the Cayley transform
$\zeta=H(z)=(1+z)/(1-z)$ that sends $1$ to $\infty$, we define a family
$\tilde{\varphi}_{s,t}:=H\circ\varphi_{s,t}\circ H^{-1}\in{\rm
Hol}(\mathbb{H})$ and assume the point $\infty$ to be the Denjoy–Wolff point
of $\tilde{\varphi}_{s,t}$. Then, the solution $\varphi_{s,t}(z)$ of the
differential equation $dw/dt=(1-w)^{2}p(w,t)$ transforms into the solution
$\tilde{\varphi}_{s,t}(\zeta)$ of $dw/dt=\tilde{p}(w,t)$ where
$\tilde{p}(\zeta,t)=2p(H^{-1}(\zeta),\,t)$.

We say that $(\tilde{\varphi}_{s,t})$ is an evolution family in $\mathbb{H}$
if $(\varphi_{s,t})$ is an evolution family in $\mathbb{D}$. Similarly, we say
that $(\tilde{f}_{t}):=(H\circ f_{t}\circ H^{-1})$ is a Loewner chain in
$\mathbb{H}$ if $(f_{t})$ is a Loewner chain in $\mathbb{D}$; and
$\tilde{p}(\cdot,t):=p(H^{-1}(\cdot),t)$ is a Herglotz function in
$\mathbb{H}$ if $p(\cdot,t)$ is a Herglotz function in $\mathbb{D}$.

######  Definition.

An evolution family $(\varphi_{s,t})$ in $\mathbb{H}$ is said to be of chordal
type, if for all $0\leq s\leq t<\infty$, all elements $\varphi_{s,t}$’s
different from ${\rm id}_{\mathbb{H}}$ share the common boundary Denjoy–Wolff
point at $\infty$. A Loewner chain $(f_{t})$ in $\mathbb{H}$ is said to be of
chordal type if the corresponding evolution family
$(\varphi_{s,t}):=(f_{t}^{-1}\circ f_{s})$ is of chordal type.

We remark that under these circumstances the half-plane version of the
Julia–Wolff–Carathéodory theorem implies that each $\varphi_{s,t}$ satisfies

${\rm Re}\,\varphi_{s,t}(z)\geq\varphi_{s,t}^{\prime}(\infty){\rm Re}\,z$

for all $z\in\mathbb{H}$, where $\varphi_{s,t}^{\prime}(\infty)$ is the
Carathéodory angular derivative of $(\varphi_{s,t})$ defined by

$\varphi_{s,t}^{\prime}(\infty):=\angle\lim_{z\to\infty}\frac{\varphi_{s,t}(z)}{z}=\frac{1}{(H^{-1}\circ\varphi_{s,t}\circ
H)^{\prime}(1)}\geq 1.$

Based on the above arguments, any chordal evolution family $(\varphi_{s,t})$
and the corresponding Loewner chain $(f_{t})$ satisfy the following chordal
Loewner–Kufarev ODE and PDE, respectively:

$\frac{d\varphi_{s,t}(z)}{dt}=p(\varphi_{s,t}(z),t),\quad\varphi_{s,s}(z)=z$
(2.10)

for all $z\in\mathbb{H}$ and almost every $t\geq s\geq 0$, and

$\dot{f}_{t}(z)=-p(z,t)f_{t}^{\prime}(z)$ (2.11)

for all $z\in\mathbb{H}$ and almost every $t\geq 0$ with some Herglotz
function $p$ determined uniquely up to a set of measure zero on the $t$-axis.
We say that $(\varphi_{s,t})$ and $(f_{t})$ are the evolution family and the
Loewner chain associated with the Herglotz function $p$, respectively.

##  3\. Boundary condition of the Schwarzian derivative

In this section, we consider the situation where the norm of the Schwarzian
derivative of a univalent analytic function $g$ on the half-plane $\mathbb{H}$
vanishes at the boundary. Differently from the case of $\mathbb{D}$, this does
not necessarily imply that $f$ is quasiconformally extendible even if
$f(\mathbb{H})$ is a Jordan domain. We show this fact. We also prove that the
vanishing condition of the norm of the Schwarzian derivative is equivalent to
that of the pre-Schwarzian derivative.

We start with the following elementary but crucial observation for the
arguments of this section.

######  Lemma 3.1.

Let $D:=D(1,1)$ be an open disk in $\mathbb{H}$ with center $1$ and radius
$1$. For any $\epsilon>0$, there exists a horodisk $D^{\prime}\;(\subset D)$
tangent to $\partial D$ at $0$ such that $\rho_{D}^{-1}(z)/({\rm
Re}\,z)\leq\epsilon$ for all $z\in D\backslash D^{\prime}$.

###### Proof.

The hyperbolic density and the distance to the boundary satisfy the estimates
$(4\rho_{D}(z))^{-1}\leq d(z,\partial D)\leq\rho_{D}^{-1}(z)$ (see [33,
p.92]). Then, we consider the ratio $d(z,\partial D)/({\rm Re}z)$ instead of
$\rho_{D}^{-1}(z)/({\rm Re}\,z)$.

For a given constant $a>1$, we define a curve

$\Gamma_{a}:=\left\\{z\in D:\;\frac{d(z,\partial D)}{{\rm
Re}z}=\frac{1}{a}\right\\}.$

By computation, $\Gamma_{a}$ is an ellipse whose points $z=x+iy$ hold an
equation

$\frac{\left(x-a/(a+1)\right)^{2}}{\left(a/(a+1)\right)^{2}}+\frac{y^{2}}{(a-1)/(a+1)}=1.$

The ellipse $\Gamma_{a}$ becomes larger as $a$ increases and tends to
$\partial D$ as $a\to\infty$. Moreover, $\Gamma_{a}$ is contained in
$\overline{D_{a}}$, where

$D_{a}:=\left\\{z\in D:\;\left|z-a/(a+1)\right|<a/(a+1)\right\\}$

is a horodisk tangent to $\partial D$ at $0$. Then, we see that $d(z,\partial
D)/({\rm Re}z)\leq 1/a$ for all $z\in D\backslash D_{a}$. Consequently, by
taking $a=4/\epsilon$ and setting $D^{\prime}=D_{4/\epsilon}$, we have that
$\rho_{D}^{-1}(z)/({\rm Re}z)\leq\epsilon$ for all $z\in D\backslash
D^{\prime}$. ∎

This lemma makes it possible to construct conformal mappings on $\mathbb{H}$
easily that satisfy the vanishing condition of their Schwarzian derivatives at
the boundary. Here is a general manner for this.

######  Proposition 3.2.

Let $g(\zeta)$ be a conformal mapping on $\mathbb{H}$. Let $D=D(1,1)$ and let
$\zeta=\varphi(z)=2/(z+1)$ be a Möbius transformation that maps $\mathbb{H}$
onto $D$ with $\varphi(\infty)=0$. Then, the conformal mapping
$f:=g\circ\varphi$ on $\mathbb{H}$ satisfies

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)^{2}|Sf(z)|=0.$ (3.1)

###### Proof.

Since $g$ is univalent on $\mathbb{H}$, we have $(2{\rm
Re}\,\zeta)^{2}|Sg(\zeta)|\leq 6$ for all $\zeta\in\mathbb{H}$ (see [27, p.
60]), and then we have that for all $\zeta\in D$,

$\begin{split}\rho_{D}^{-2}(\zeta)|Sg(\zeta)|&=(2{\rm
Re}\,\zeta)^{2}|Sg(\zeta)|\frac{\rho_{D}^{-2}(\zeta)}{(2{\rm
Re}\,\zeta)^{2}}\leq\frac{3}{2}\left(\frac{\rho_{D}^{-1}(\zeta)}{{\rm
Re}\,\zeta}\right)^{2}.\end{split}$

According to Lemma 3.1, for any $\epsilon>0$, there exists a horodisk
$D^{\prime}\;(\subset D)$ tangent to $\partial D$ at $0$ such that
$\rho_{D}^{-2}(\zeta)|Sg(\zeta)|\leq\epsilon$ for all $\zeta\in D\backslash
D^{\prime}$.

Set $H_{\epsilon}=\varphi^{-1}(D^{\prime})$, which is a half-plane in
$\mathbb{H}$. As

$(2{\rm Re}z)^{2}|Sf(z)|=\rho_{D}^{-2}(\zeta)|Sg(\zeta)|$

for all $z\in\mathbb{H}$ by the invariance under the Möbius transformation
$\varphi$, we conclude that for all $z\in\mathbb{H}\backslash H_{\epsilon}$,
$(2{\rm Re}z)^{2}|Sf(z)|\leq\epsilon$. ∎

As a particular application of this construction, we obtain a counter-example
to the statement of Theorem 1.2 when we drop the extra assumptions $\bf(A)$
and $\bf(B)$.

######  Corollary 3.3.

There exists a conformal mapping $f$ on $\mathbb{H}$ satisfying both

  1. (i)

the image $f(\mathbb{H})$ is a Jordan domain, but not a quasidisk; and

  2. (ii)

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)^{2}|Sf(z)|=0$, that is, (3.1).

###### Proof.

Consider a half horizontal parallel strip $V$ in $\mathbb{H}$ with vertices
$\pm i\pi/2$, that is,

$V:=\\{w=u+iv\in\mathbb{H}:-\pi/2<v<\pi/2\\}.$

By the Schwarz–Christoffel formula,

$w=g(\zeta):=-i\arcsin\left(\frac{i}{\zeta}\right)=-\log\left(-\frac{1}{\zeta}+\sqrt{1+\frac{1}{\zeta^{2}}}\right)$

maps $\mathbb{H}$ conformally onto $V$ with $g(0)=\infty$, $g(i)=-i\pi/2$ and
$g(-i)=i\pi/2$, where we choose the branch of $\arcsin(i/\zeta)$ so that it
takes $\pi/2$ at $\zeta=i$. Following this, we see the image of $D=D(1,1)$
under $g$ is a Jordan domain, but not a quasidisk since its boundary has a
cusp at $\infty$.

Let $\varphi$ be the Möbius transformation as in Proposition 3.2. We define

$f(z):=g\circ\varphi(z)=-i\arcsin\left(i(z+1)/2\right).$

Then, $f(\mathbb{H})=g(D)$ satisfies condition (i). Moreover, we see that the
conformal mapping $f$ satisfies condition (ii) by Proposition 3.2. ∎

In the latter half of this section, we compare the boundary behavior of
Schwarzian derivatives and pre-Schwarzian derivatives for univalent analytic
functions on $\mathbb{H}$. This will be used in the proof of Theorem 1.2
later.

First, we state a claim below obtained from the Cauchy integral formula. As in
[23, Lemma 4], if $h$ is an analytic function on a domain
$D\subset\mathbb{C}$, then

$\sup_{z\in D}|h^{\prime}(z)|d(z,\partial D)^{2}\leq 4\sup_{z\in
D}|h(z)|d(z,\partial D).$ (3.2)

This yields the implication

$\lim_{z\to\partial D}|h(z)|d(z,\partial
D)=0\quad\Longrightarrow\quad\lim_{z\to\partial D}|h^{\prime}(z)|d(z,\partial
D)^{2}=0.$ (3.3)

Indeed, by setting $r:=d(z,\partial D)/2$ for a fixed $z\in D$, (3.3) easily
follows from:

$\begin{split}|h^{\prime}(z)|&=\left|\frac{1}{2\pi
i}\int_{|\zeta-z|=r}\frac{h(\zeta)}{(\zeta-z)^{2}}d\zeta\right|\leq\frac{1}{r}\sup_{|\zeta-z|=r}|h(\zeta)|\\\
&\leq\frac{2}{d(z,\partial
D)}\sup_{|\zeta-z|=r}\left(|h(\zeta)|d(\zeta,\partial
D)\right)\sup_{|\zeta-z|=r}\frac{1}{d(\zeta,\partial D)}\\\
&=\frac{4}{d(z,\partial
D)^{2}}\sup_{|\zeta-z|=r}\left(|h(\zeta)|d(\zeta,\partial
D)\right).\end{split}$

Suppose that $D$ is a simply connected domain with the hyperbolic density
$\rho_{D}$. Since $\rho_{D}(z)\leq 1/d(z,\partial D)\leq 4\rho_{D}(z)$, it
follows from (3.2) and (3.3) that

$\sup_{z\in D}|h^{\prime}(z)|\rho_{D}^{-2}(z)\leq 64\sup_{z\in
D}|h(z)|\rho_{D}^{-1}(z);$ $\lim_{z\to\partial
D}|h(z)|\rho_{D}^{-1}(z)=0\quad\Longrightarrow\quad\lim_{z\to\partial
D}|h^{\prime}(z)|\rho_{D}^{-2}(z)=0.$

For an analytic and locally univalent function $g$ on $D$, we can apply these
facts to the pre-Schwarzian derivative $Pg$ for $g$ and the Schwarzian
derivative $Sg=(Pg)^{\prime}-(Pg)^{2}/2$ to obtain the following implications:

$\begin{split}\sup_{z\in
D}|Pg(z)|\rho_{D}^{-1}(z)=:\lambda\quad&\Longrightarrow\quad\sup_{z\in
D}|Sg(z)|\rho_{D}^{-2}(z)\leq 64\lambda+\lambda^{2}/2;\\\ \lim_{z\to\partial
D}|Pg(z)|\rho_{D}^{-1}(z)=0\quad&\Longrightarrow\quad\lim_{z\to\partial
D}|Sg(z)|\rho_{D}^{-2}(z)=0.\end{split}$ (3.4)

Next, we consider the converse implication. The following result is
essentially contained in Becker [11].

######  Proposition 3.4.

Let $f$ be a univalent analytic function on $\mathbb{D}$ and let
$f(\mathbb{D})$ be a Jordan domain. Suppose further that $f$ satisfies
condition (1.4). For $t\in(0,1)$, set

$\begin{split}\hat{\beta}(t)&:=\sup_{1-t\leq|z|<1}(1-|z|^{2})|Pf(z)|,\\\
\hat{\sigma}(t)&:=\sup_{1-t\leq|z|<1}(1-|z|^{2})^{2}|Sf(z)|.\end{split}$

Then, there is some $t_{0}\in(0,1)$ such that

$\hat{\beta}(t^{1+\epsilon})\leq 4\hat{\sigma}(t)+8t^{\epsilon}$

for $0<t\leq t_{0}$. Here, $\epsilon>0$ can take an arbitrary positive
constant.

###### Proof.

Suppose (1.4) holds. Then by Theorem 1.1, $f$ has a quasiconformal extension
to $\mathbb{C}$ with complex dilatation $\mu$ satisfying

$\hat{k}(t):=\\{\|\mu(z)\|_{\infty}:\,1<|z|\leq 1+t\\}\leq\hat{\sigma}(t)/2$
(3.5)

for any $0<t\leq t_{0}$, where $t_{0}<1$ is some constant. By applying Lehto’s
majorization method, [11, Theorem 2] implies that

$\hat{\beta}(t^{1+\epsilon})\leq 8(\hat{k}(t)+t^{\epsilon}),\quad\epsilon>0$
(3.6)

for any $0<t<1$. Combining (3.5) and (3.6), we obtain the assertion. ∎

By the combination of this proposition with Lemma 3.1, we can obtain the
equivalence between the vanishing conditions of the norms for the Schwarzian
derivative and the pre-Schwarzian derivative.

######  Theorem 3.5.

Let $f$ be a univalent analytic function on $\mathbb{H}$. For $t>0$, set

$\begin{split}\beta(t)&:=\sup_{0<{\rm Re}z\leq t}(2{\rm Re}z)|Pf(z)|,\\\
\sigma(t)&:=\sup_{0<{\rm Re}z\leq t}(2{\rm Re}z)^{2}|Sf(z)|.\end{split}$

Then, $\beta(t)\to 0$ if and only if $\sigma(t)\to 0$ as $t\to 0$.

###### Proof.

If $\beta(t)\to 0$ as $t\to 0$, then $\sigma(t)\to 0$ as $t\to 0$ by (3.4). We
will show the opposite implication by means of of Proposition 3.4.

Now suppose $\sigma(t)\to 0$ as $t\to 0$. We choose $t_{0}\in(0,1)$ such that
$\sigma(t_{0})\leq 2k\,(<2)$. For an arbitrary boundary point $iy_{0}\in
i\mathbb{R}$, take the open disk $D:=D(1+iy_{0},1)\subset\mathbb{H}$ of radius
$1$ that is tangent to $i\mathbb{R}$ at $iy_{0}$. For any $z$ in the domain
$\\{z\in D:0<{\rm Re}\,z\leq t_{0}\\}$, obviously we have

$\rho_{D}^{-2}(z)|Sf(z)|\leq(2{\rm Rez})^{2}|Sf(z)|\leq\sigma(t_{0}).$ (3.7)

Moreover, by Lemma 3.1, there exists a horodisk $D^{\prime}\,(\subset D)$
tangent to $\partial D$ at $iy_{0}$ such that $\rho_{D}^{-1}(z)/({\rm
Re}\,z)\leq(2\sigma(t_{0})/3)^{1/2}$ for all $z\in D\backslash D^{\prime}$.
This yields that

$\begin{split}\rho_{D}^{-2}(z)|Sf(z)|&=(2{\rm
Re}z)^{2}|Sf(z)|\left(\frac{\rho_{D}^{-1}(z)}{2{\rm Re}z}\right)^{2}\leq
6\left(\frac{\rho_{D}^{-1}(z)}{2{\rm
Re}z}\right)^{2}\leq\sigma(t_{0})\end{split}$ (3.8)

for all $z\in D\backslash D^{\prime}$.

Combining (3.7) and (3.8), we can define a function
$t_{1}:=\lambda(t_{0})\,(\leq t_{0})$ tending to $0$ monotonically and
continuously as $t_{0}\to 0$ such that

$\hat{\sigma}_{D}(t_{1}):=\sup_{0<d(z,\partial D)\leq t_{1}\atop z\in
D}\rho_{D}^{-2}(z)|Sf(z)|\leq\sigma(t_{0}).$ (3.9)

Similarly, we define

$\hat{\beta}_{D}(t_{1}):=\sup_{0<d(z,\partial D)\leq t_{1}\atop z\in
D}\rho_{D}^{-1}(z)|Pf(z)|.$

By Proposition 3.4 for $\epsilon=1$, we have

$\hat{\beta}_{D}(t_{1}^{2})\leq 4\hat{\sigma}_{D}(t_{1})+8t_{1}$ (3.10)

for all sufficiently small $t_{1}>0$.

Now we consider the estimate only on the particular segment $z=x+iy_{0}$ with
$0<x<t_{1}$ in the annulus $\\{z\in D:0<d(z,\partial D)<t_{1}\\}$. On this
segment, we have that

$({\rm Re}z)|Pf(z)|=d(z,\partial
D)|Pf(z)|\leq\rho_{D}^{-1}(z)|Pf(z)|\leq\hat{\beta}_{D}(t_{1}).$ (3.11)

Since $iy_{0}\in i\mathbb{R}$ is taken arbitrarily, combining (3.9), (3.10),
and (3.11) together, we obtain

$\beta(t_{1}^{2})\leq 8\sigma(\lambda^{-1}(t_{1}))+16t_{1}.$

This tends to $0$ as $t_{1}\to 0$. ∎

######  Remark.

To the best of our knowledge, the equivalence of the conditions in Theorem 3.5
is known only under the extra assumption that $f$ can be quasiconformally
extended to $\overline{\mathbb{C}}$ (see [35, Theorem 6.2], [38, Theorem
5.1]). By a similar proof or by an application, we see the statement of
Theorem 3.5 is true also for $\mathbb{D}$. Namely, $\hat{\beta}(t)\to 0$ if
and only if $\hat{\sigma}(t)\to 0$ as $t\to 0$ without any extra assumption on
a univalent analytic function $g$ on $\mathbb{D}$. This improves the result in
[11].

##  4\.  Quasiconformal extensions by means of Schwarzian derivative (Proof of
Theorem 1.2)

In this section, we prove Theorem 1.2. For this argument, the following
chordal analogue of Becker’s result [8] mentioned in the introduction plays an
important role, which was obtained in Gumenyuk and Hotta [22, Theorem 3.5].

######  Theorem 4.1.

Let $(f_{t})$ be a chordal Loewner chain over the interval $0\leq t<\tau$ in
$\mathbb{H}$ with Herglotz function $p$. Suppose that there exists a constant
$k\in[0,1)$ such that

$p(z,t)\in U(k)=\left\\{w\in\mathbb{H}:\;\left|\frac{w-1}{w+1}\right|\leq
k\right\\}$ (4.1)

for all $z\in\mathbb{H}$ and almost every $0\leq t<\tau$. Then,

  1. (i)

$f_{t}$ has a continuous extension to $i\mathbb{R}$ for all $0\leq t<\tau$;

  2. (ii)

$f_{t}$ can be $k$-quasiconformally extended to $\\{z\in\mathbb{C}:0\leq-{\rm
Re}\,z<\tau-t\\}$ by setting $f_{t}(z)=f_{t-{\rm Re}z}(i\,{\rm Im}z)$ for all
$0\leq t<\tau$;

  3. (iii)

all elements of the evolution family $(\varphi_{s,t})$ associated with $p$ are
$k$-quasiconformally extendible to $\overline{\mathbb{C}}$ with
$\varphi_{s,t}(\infty)=\infty$ for all $s\geq 0$ and $s\leq t<\tau$.

We remark that the original statement of [22, Theorem 3.5] was concerning the
case of $\tau=+\infty$, namely, the time variable $t$ extends to $+\infty$ as
usual in the Loewner theory. By examining its proof, however, we see that the
above form of the statement is valid for any constant $\tau$ in the interval
$(0,+\infty]$.

For the construction of the quasiconformal extension $\hat{h}$ of $h$ in
Theorem 1.2, a chordal Loewner chain $(h_{t})$ over the interval $[0,\tau)$
will be formed so that $h_{0}(z)=h(z)$ and the boundary values of $h_{t}(z)$
yield the extension $\hat{h}$, as is shown in (ii) of Theorem 4.1. In the next
lemma, we show that a canonical family of analytic functions $(h_{t})$ can be
constructed as in [22, Proposition 5.5] and it satisfies the chordal
Loewner–Kufarev PDE with an appropriate Herglotz function.

######  Lemma 4.2.

Let $h$ be univalent and analytic in $\mathbb{H}$ with
$\lim_{z\to\infty}h(z)=\infty$. Let $h$ satisfy

$\lim_{{\rm Re}z\to 0^{+}}\frac{1}{2}(2{\rm Re}z)^{2}\left|Sh(z)\right|=0.$
(1.5)

Then, for any $0<k<1$, there is a positive constant $\tau_{0}>0$ such that
functions on $\mathbb{H}$ defined by

$h_{t}(z)=h(z+t)-\frac{2th^{\prime}(z+t)}{1+tPh(z+t)}$ (4.2)

for $0\leq t\leq\tau_{0}$ are analytic, and this family $(h_{t})$ satisfies
$h_{0}=h$ and the chordal Loewner–Kufarev PDE

$\dot{h}_{t}(z)=-p(z,t)h^{\prime}_{t}(z)$ (4.3)

with Herglotz function $p(z,t)$ satisfying (4.1).

###### Proof.

In view of Theorem 3.5, (1.5) implies that

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)\left|Ph(z)\right|=0,$ (1.8)

that is, for any $0<k<1$, there exists a positive constant $t_{0}$ such that

$(2{\rm Re}z)\left|Ph(z)\right|\leq k$

for all $z\in\mathbb{H}$ with $0<{\rm Re}\,z\leq t_{0}$. It follows that

$t|Ph(iy+t)|={\rm Re}(iy+t)|Ph(iy+t)|\leq k/2$

for all $iy\in i\mathbb{R}$ and all $0<t\leq t_{0}$. A generalized maximal
principle (see [20, p.56]) implies that $tPh(z+t)$, as an analytic function of
the variable $z$, is bounded in $\mathbb{H}$ such that

$t|Ph(z+t)|\leq k/2,\qquad 0\leq t\leq t_{0}.$ (4.4)

Similarly, according to (1.5), there exists a positive constant
$\tau_{0}\;(<t_{0})$ such that

$\frac{1}{2}(2{\rm Re}z)^{2}\left|Sh(z)\right|\leq k,$

for all $z\in\mathbb{H}$ with $0<{\rm Re}\,z\leq\tau_{0}$. From this it
follows that

$2t^{2}|Sh(iy+t)|=\frac{1}{2}\left(2{\rm
Re}(iy+t)\right)^{2}\left|Sh(iy+t)\right|\leq k$

for all $iy\in i\mathbb{R}$ and all $0<t\leq\tau_{0}$. The generalized maximal
principle again implies that

$2t^{2}|Sh(z+t)|\leq k,\qquad 0\leq t\leq\tau_{0}$ (4.5)

for all $z\in\mathbb{H}$.

We consider the family of functions given by (4.2). This is well-defined to be
analytic in $\mathbb{H}$ for $0\leq t\leq t_{0}$ by (4.4). Obviously,
$h_{0}(z)=h(z)$. By computation, we have

$\dot{h}_{t}(z):=\frac{dh_{t}(z)}{dt}=-h^{\prime}(z+t)\frac{1-2t^{2}Sh(z+t)}{\left(1+tPh(z+t)\right)^{2}},$
(4.6)

and

$h^{\prime}_{t}(z):=\frac{dh_{t}(z)}{dz}=h^{\prime}(z+t)\frac{1+2t^{2}Sh(z+t)}{\left(1+tPh(z+t)\right)^{2}}.$
(4.7)

From these expressions (4.6) and (4.7), it follows that $(h_{t})$ satisfies
the chordal Loewner–Kufarev PDE (4.3) with Herglotz function

$p(z,t)=\frac{1-2t^{2}Sh(z+t)}{1+2t^{2}Sh(z+t)}.$ (4.8)

Moreover, by (4.5) we can deduce that

$\left|\frac{p(z,t)-1}{p(z,t)+1}\right|=2t^{2}|Sh(z+t)|\leq k<1$ (4.9)

for all $z\in\mathbb{H}$ and all $0\leq t\leq\tau_{0}$. Namely, $p(z,t)\in
U(k)$. ∎

In the rest of this section, we will focus our attention on showing the
univalence of the analytic function $h_{t}$ in $\mathbb{H}$ defined by (4.2)
for each $t$ in some interval. The arguments will be separated into the cases
of assumptions $\mathbf{(A)}$ and $\mathbf{(B)}$.

######  Theorem 4.3.

Let $h$ be univalent and analytic in $\mathbb{H}$ with
$\lim_{z\to\infty}h(z)=\infty$ and let $h(\mathbb{H})$ be a Jordan domain. Let
$h$ satisfy

$\lim_{{\rm Re}z\to 0^{+}}\frac{1}{2}(2{\rm Re}z)^{2}\left|Sh(z)\right|=0.$
(1.5)

Suppose $\mathbf{(A)}$: both $h$ and $h^{-1}$ are locally uniformly continuous
in a neighborhood of $\infty$. Then, there is a positive constant
$\tau\leq\tau_{0}$ such that the analytic function $h_{t}$ on $\mathbb{H}$
defined by (4.2) is univalent for all $0\leq t\leq\tau$.

###### Proof.

The proof contains several claims. For all claims as well as other
explanations, the circumstances of the theorem are always assumed. By Lemma
4.2, the family $(h_{t})$ defined by (4.2) for $0\leq t\leq\tau_{0}$ satisfies
the chordal Loewner–Kufarev PDE (4.3) with the Herglotz function $p(z,t)$ in
(4.1). In particular, $|p(z,t)|\leq K$ for $K:=(1+k)/(1-k)$.

Let $(\varphi_{s,t})$ be the evolution family associated with this Herglotz
function $p$ for all $0\leq s\leq t\leq\tau_{0}$. Namely, $(\varphi_{s,t})$
satisfies the chordal Loewner–Kufarev ODE:

$\dot{\varphi}_{s,t}(z)=p(\varphi_{s,t}(z),t),\quad 0\leq s\leq
t\leq\tau_{0},\quad\varphi_{s,s}(z)=z.$ (4.10)

By Theorem 4.1, $\varphi_{s,t}$ is $k$-quasiconformally extendible to
$\overline{\mathbb{C}}$ with $\varphi_{s,t}(\infty)=\infty$ for some $0<k<1$.
Combining (4.3) and (4.10), we obtain that for any fixed $z\in\mathbb{H}$,

$\begin{split}\frac{d}{dt}(h_{t}\circ\varphi_{s,t}(z))&=h_{t}^{\prime}(\varphi_{s,t}(z))\dot{\varphi}_{s,t}(z)+\dot{h}_{t}(\varphi_{s,t}(z))\\\
&=h_{t}^{\prime}(\varphi_{s,t}(z))\dot{\varphi}_{s,t}(z)-p(\varphi_{s,t}(z),t)h_{t}^{\prime}(\varphi_{s,t}(z))\\\
&=h_{t}^{\prime}(\varphi_{s,t}(z))(\dot{\varphi}_{s,t}(z)-p(\varphi_{s,t}(z),t))=0.\end{split}$

Thus, $h_{t}\circ\varphi_{s,t}(z)$ does not depend on $t$. By taking $t=s$ we
see that $h_{t}\circ\varphi_{s,t}(z)=h_{s}(z)$. The equations
$\varphi_{0,t}=\varphi_{s,t}\circ\varphi_{0,s}$ and
$h_{s}=h_{t}\circ\varphi_{s,t}$ will be frequently used below.

We note that the univalent analytic function $h$ on $\mathbb{H}$ has a
continuous and injective extension to
$i\hat{\mathbb{R}}=i\mathbb{R}\cup\\{\infty\\}$ with $h(\infty)=\infty$ by the
Carathéodory theorem (see [33, p.18]) since $h(\mathbb{H})$ is a Jordan
domain.

######  Claim 0.

The value $h_{t}(z)$ converges to $h(z_{0})$ as $z\in\mathbb{H}$ tends to
$z_{0}\in\mathbb{H}\cup i\hat{\mathbb{R}}$ and $t\in[0,\tau_{0}]$ tends to
$0$.

###### Proof.

For any $z\in\mathbb{H}$ and $0\leq t\leq t^{\prime}\leq\tau_{0}$,

$\begin{split}|\varphi_{t,t^{\prime}}(z)-z|&=|\varphi_{t,t^{\prime}}(z)-\varphi_{t,t}(z)|\\\
&=\left|\int_{t}^{t^{\prime}}\dot{\varphi}_{t,s}(z)ds\right|=\left|\int_{t}^{t^{\prime}}p(\varphi_{t,s}(z),s)ds\right|\\\
&\leq\int_{t}^{t^{\prime}}|p(\varphi_{t,s}(z),s)|ds\leq
K(t^{\prime}-t)\end{split}$ (4.11)

is satisfied. This is also true for $z\in i\mathbb{R}$ since each
$\varphi_{t,t^{\prime}}$ has the continuous extension to $i\mathbb{R}$. In
particular,

$|\varphi_{0,t}(z)-z|\leq Kt$ (4.12)

for any $z\in\mathbb{H}\cup i\mathbb{R}$ and $0\leq t\leq\tau_{0}$. Let
$\Omega(M):=\\{z\in\mathbb{H}:|z|\leq M\\}$ for any $M>0$. Then,

$|\varphi_{t,\tau_{0}}(z)|\leq|\varphi_{t,\tau_{0}}(z)-z|+|z|\leq K\tau_{0}+M$
(4.13)

for any $z\in\Omega(M)$. This shows that the family
$\\{\varphi_{t,\tau_{0}}\\}_{t\in[0,\tau_{0}]}$ is uniformly bounded on the
compact set $\overline{\Omega}(M)=\\{z\in\mathbb{H}\cup i\mathbb{R}:|z|\leq
M\\}$ for any $M>0$.

If the family of $k$-quasiconformal mappings
$\varphi_{t,\tau_{0}}:\mathbb{C}\to\mathbb{C}$ is uniformly bounded on
$\overline{\Omega}(M+K\tau_{0})$, then it is also equi-continuous on
$\overline{\Omega}(M+K\tau_{0})$ (see [28, Theorem II.4.1]), and thus
uniformly Hölder continuous with exponent $1/K$ and with multiplicative
constant $C_{1}>0$ (see [28, Theorem II.4.3]). From this and (4.12), it
follows that

$\begin{split}|\varphi_{t,\tau_{0}}(z)-\varphi_{0,\tau_{0}}(z)|&=|\varphi_{t,\tau_{0}}(z)-\varphi_{t,\tau_{0}}\circ\varphi_{0,t}(z)|\\\
&\leq C_{1}|z-\varphi_{0,t}(z)|^{1/K}\leq C_{1}(Kt)^{1/K}\end{split}$ (4.14)

for any $z\in\Omega(M)$.

Since $p(z,t)\in U(k)$ for all $z\in\mathbb{H}$ and all $0\leq t\leq\tau_{0}$,
there exists a positive constant $C_{2}>0$ such that ${\rm Re}\,p(z,t)\geq
C_{2}$. It follows that

$\begin{split}{\rm Re}\,(\varphi_{0,t}(z))-{\rm Re}\,z&=\int_{0}^{t}{\rm
Re}\,(\dot{\varphi}_{0,s}(z))ds\\\ &=\int_{0}^{t}{\rm
Re}\,(p(\varphi_{0,s}(z),s))ds\geq C_{2}t.\end{split}$

This implies that $\varphi_{0,\tau_{0}}(\mathbb{H})$ is contained in
$\\{z\in\mathbb{H}:{\rm Re}\,z>C_{2}\tau_{0}\\}$. Combining this with (4.13)
and (4.14), we can take a compact convex subset $W$ in $\mathbb{H}$ such that
$\varphi_{t,\tau_{0}}(z)$ and $\varphi_{0,\tau_{0}}(z)$ are contained in $W$
for all $z\in\Omega(M)$ and for all sufficiently small $t$.

Let $C_{3}:=\max_{w\in W}|h_{\tau_{0}}^{\prime}(w)|<+\infty$. Then, for any
two points $w_{1}$, $w_{2}\in W$,

$\begin{split}|h_{\tau_{0}}(w_{1})-h_{\tau_{0}}(w_{2})|&=\left|\int_{0}^{1}(h_{\tau_{0}}((1-t)w_{1}+tw_{2}))_{t}^{\prime}dt\right|\\\
&=|w_{2}-w_{1}|\left|\int_{0}^{1}h_{\tau_{0}}^{\prime}((1-t)w_{1}+tw_{2})dt\right|\leq
C_{3}|w_{2}-w_{1}|,\end{split}$

from which we deduce that

$\begin{split}|h_{t}(z)-h(z_{0})|&\leq|h_{t}(z)-h(z)|+|h(z)-h(z_{0})|\\\
&=|h_{\tau_{0}}\circ\varphi_{t,\tau_{0}}(z)-h_{\tau_{0}}\circ\varphi_{0,\tau_{0}}(z)|+|h(z)-h(z_{0})|\\\
&\leq
C_{3}|\varphi_{t,\tau_{0}}(z)-\varphi_{0,\tau_{0}}(z)|+|h(z)-h(z_{0})|\end{split}$
(4.15)

if $z\in\Omega(M)$. In the case that $z_{0}\neq\infty$, by choosing
$\Omega(M)$ according to $z_{0}$, we see that $h_{t}(z)\to h(z_{0})$ as $z\to
z_{0}$ and $t\to 0$ by (4.14) and (4.15).

Finally, we assume that $z_{0}=\infty$. By the equation
$h_{\tau_{0}}\circ\varphi_{0,\tau_{0}}=h$, we have
$\lim_{z\to\infty}h_{\tau_{0}}(z)=\lim_{z\to\infty}h\circ\varphi_{0,\tau_{0}}^{-1}(z)=\infty$.
Moreover,

$\begin{split}|\varphi_{t,\tau_{0}}(z)|&\geq|z|-|\varphi_{t,\tau_{0}}(z)-\varphi_{t,t}(z)|\\\
&=|z|-\left|\int_{t}^{\tau_{0}}\dot{\varphi}_{t,s}(z)ds\right|\\\
&\geq|z|-\int_{t}^{\tau_{0}}|p(\varphi_{t,s}(z),s)|ds\geq|z|-K\tau_{0}.\end{split}$

Hence,
$h_{t}(z)=h_{\tau_{0}}\circ\varphi_{t,\tau_{0}}(z)\to\infty\,(=h(\infty))$ as
$z\to\infty$ (independent of $t$). ∎

######  Claim 1.

There exist positive constants $\tau_{1}\;(\leq\tau_{0})$ and $r$ such that
for any $0\leq t\leq\tau_{1}$, $h_{t}(z)$ is univalent on each square

$Q_{2r}:=(0,2r)\times(y_{0}-r,y_{0}+r)\subset\mathbb{H},$

where $y_{0}\in\mathbb{R}$ can be chosen arbitrarily.

Remark that the square $Q_{2r}$, as a quasidisk, satisfies the Schwarzian
univalence criterion (see [23, Corollary 5], [27, p.126]). Precisely, if a
function $g$ is locally univalent and analytic in $Q_{2r}$ and if

$\sup_{z\in Q_{2r}}|Sg(z)|\rho_{Q_{2r}}^{-2}(z)\leq 1/2,$

then $g$ is univalent in $Q_{2r}$. Moreover, by (3.4), there exists a positive
constant $a$ such that if

$\sup_{z\in Q_{2r}}|Pg(z)|\rho_{Q_{2r}}^{-1}(z)\leq a,$

then $g$ is univalent in $Q_{2r}$.

We also need the following result which can be deduced easily from the proof
of [35, Lemma 6.3].

######  Proposition 4.4.

Let $\psi$ be an analytic function of $z=x+iy$ in $\mathbb{H}$ satisfying a
condition $\lim_{x\to+\infty}\psi(x+iy)=0$ uniformly for $y\in\mathbb{R}$.
Then, for any constant $\alpha>0$,

$\sup_{z\in\mathbb{H}}|\psi(z)|x^{\alpha}<\infty\quad\Longleftrightarrow\quad\sup_{z\in\mathbb{H}}|\psi^{\prime}(z)|x^{\alpha+1}<\infty,$

and both terms are comparable with comparison constants depending only on
$\alpha$. Moreover,

$\lim_{x\to 0^{+}}|\psi(z)|x^{\alpha}=0\quad\Longleftrightarrow\quad\lim_{x\to
0^{+}}|\psi^{\prime}(z)|x^{\alpha+1}=0.$

###### Proof of Claim 1.

By (4.5) and (4.7), we see that $h^{\prime}_{t}(z)\neq 0$ for all
$z\in\mathbb{H}$ and all $0\leq t\leq\tau_{0}$. A direct computation yields
that

$\begin{split}Ph_{t}(z)&:=\frac{h_{t}^{\prime\prime}(z)}{h_{t}^{\prime}(z)}=(\log
h_{t}^{\prime}(z))^{\prime}\\\
&=Ph(z+t)+\frac{2t^{2}\left(Sh(z+t)\right)^{\prime}}{1+2t^{2}Sh(z+t)}-\frac{2t\left(Ph(z+t)\right)^{\prime}}{1+tPh(z+t)}.\end{split}$

From this we have

$\begin{split}(2{\rm Re}z)\left|Ph_{t}(z)\right|&\leq(2{\rm
Re}(z+t))|Ph(z+t)|\\\ &\quad+\frac{(2{\rm
Re}(z+t))^{3}|(Sh(z+t))^{\prime}|}{1-(2{\rm
Re}(z+t))^{2}|Sh(z+t)|}+\frac{(2{\rm
Re}(z+t))^{2}|(Ph(z+t))^{\prime}|}{1-(2{\rm Re}(z+t))|Ph(z+t)|}.\end{split}$

Then, applying (1.5) and (1.8), we conclude by Proposition 4.4 that there
exists a positive constant $\tau_{1}\;(\leq\tau_{0})$ such that

$(2{\rm Re}z)\left|Ph_{t}(z)\right|\leq a$

for any $z\in\mathbb{H}$ and $t\geq 0$ with $0<{\rm Re}(z+t)\leq 3\tau_{1}$.
In particular, it holds for all $z\in\mathbb{H}$ with $0<{\rm Re}\,z\leq 2r$
and for all $0\leq t\leq\tau_{1}$ by taking $r=\tau_{1}$. We fix this constant
$r>0$.

By the monotone principle of hyperbolic densities with respect to the
inclusion of domains (see [27, p.6]), we have

$\rho_{Q_{2r}}^{-1}(z)\left|Ph_{t}(z)\right|\leq(2{\rm
Re}z)\left|Ph_{t}(z)\right|\leq a$

for all $z\in Q_{2r}$, which implies that $h_{t}$ is univalent in each
$Q_{2r}$ for any $0\leq t\leq\tau_{1}$. ∎

######  Claim 2.

There exists a positive constant $\tau_{2}\;(\leq\tau_{0})$ such that for each
$0\leq t\leq\tau_{2}$, $h_{t}$ is univalent in the half-plane
$\mathbb{H}_{r}:=\\{z\in\mathbb{H}:{\rm Re}z>r\\},$ where $r$ is the constant
chosen in Claim 1.

###### Proof.

Since $h$ and $\varphi_{0,t}$ are univalent in $\mathbb{H}$ for any $0\leq
t\leq\tau_{0}$ and since $h_{t}\circ\varphi_{0,t}=h$, we see that $h_{t}$ is
univalent in the domain $\varphi_{0,t}(\mathbb{H})$. Here, (4.11) in
particular implies that $\varphi_{0,t}$ converges to ${\rm id}_{\mathbb{H}}$
uniformly with respect to $z\in\mathbb{H}$ as $t\to 0$. Then, there exists a
positive constant $\tau_{2}\;(\leq\tau_{0})$ such that
$\mathbb{H}_{r}\subset\varphi_{0,t}(\mathbb{H})$ for each $0\leq
t\leq\tau_{2}$, and thus $h_{t}$ is univalent in $\mathbb{H}_{r}$. ∎

Let us recall the well-known Koebe distortion theorem (see [33, Theorem 1.3]).
It says that if $g$ is analytic and univalent in $\mathbb{D}$ then for all
$z\in\mathbb{D}$ it holds that

$|g^{\prime}(0)|\frac{|z|}{(1+|z|)^{2}}\leq|g(z)-g(0)|\leq|g^{\prime}(0)|\frac{|z|}{(1-|z|)^{2}}.$

Applying it to an analytic and univalent function $g$ on a closed disk
$\overline{D}(z_{0}+t,t)$ with center $z_{0}+t$ and radius $t$ by translation
and dilation, we have that for all $z\in\overline{\mathbb{D}}$,

$t|g^{\prime}(z_{0}+t)|\frac{|z|}{(1+|z|)^{2}}\leq|g(tz+(z_{0}+t))-g(z_{0}+t)|\leq
t|g^{\prime}(z_{0}+t)|\frac{|z|}{(1-|z|)^{2}}.$

In particular, the left inequality for $z=-1$ yields that

$|g(z_{0})-g(z_{0}+t)|\geq\frac{t}{4}|g^{\prime}(z_{0}+t)|.$ (4.16)

######  Claim 3.

Suppose that assumption $\mathbf{(A)}$ holds. Namely, $h$ and $h^{-1}$ are
uniformly continuous in $\Omega_{M}:=\\{z\in\mathbb{H}:|z|>M\\}$ for some
$M>0$. Then, there exist positive constants $\tau_{3}\;(<\tau_{0})$ and $R>M$
such that if $z_{1},z_{2}\in\Omega_{R}$ with $|z_{1}-z_{2}|\geq r$, then
$h_{t}(z_{1})\neq h_{t}(z_{2})$ for all $0\leq t\leq\tau_{3}$, where $r$ is
the constant chosen in Claim 1.

###### Proof.

Let $\zeta_{1}=h(z_{1})$ and $\zeta_{2}=h(z_{2})$. As $h(\infty)=\infty$,
there is a constant $R>M$ such that if $z_{1},z_{2}\in\Omega_{R}$ then
$\zeta_{1},\zeta_{2}\in\Omega_{M}$. Since $h^{-1}$ is uniformly continuous in
$\Omega_{M}$, for the given constant $r$, there exists some $\delta>0$ such
that

$|\zeta_{1}-\zeta_{2}|<\delta\quad\Longrightarrow\quad|z_{1}-z_{2}|<r,$

or equivalently,

$|z_{1}-z_{2}|\geq r\quad\Longrightarrow\quad|\zeta_{1}-\zeta_{2}|\geq\delta$
(4.17)

for $z_{1},z_{2}\in\Omega_{R}$. Moreover, since $h$ is uniformly continuous in
$\Omega_{R}\subset\Omega_{M}$, for this $\delta$, there exists a positive
constant $\tau_{3}\;(<\tau_{0})$ such that

$|z_{1}-z_{2}|\leq\tau_{3}\quad\Longrightarrow\quad|\zeta_{1}-\zeta_{2}|\leq\delta/64$
(4.18)

for $z_{1},z_{2}\in\Omega_{R}$.

Assume that $z_{1},z_{2}\in\Omega_{R}$ with $|z_{1}-z_{2}|\geq r$ and $0\leq
t\leq\tau_{3}$. By using (4.4), (4.16), (4.17), and (4.18) in this order, we
have

$\begin{split}|h_{t}(z_{1})-h_{t}(z_{2})|&=\left|h(z_{1}+t)-h(z_{2}+t)-\frac{2th^{\prime}(z_{1}+t)}{1+tPh(z_{1}+t)}+\frac{2th^{\prime}(z_{2}+t)}{1+tPh(z_{2}+t)}\right|\\\
&\geq|h(z_{1}+t)-h(z_{2}+t)|-4t|h^{\prime}(z_{1}+t)|-4t|h^{\prime}(z_{2}+t)|\\\
&\geq|h(z_{1}+t)-h(z_{2}+t)|-16|h(z_{1}+t)-h(z_{1})|-16|h(z_{2}+t)-h(z_{2})|\\\
&\geq\delta-\delta/4-\delta/4>0.\end{split}$

This proves that $h_{t}(z_{1})\neq h_{t}(z_{2})$. ∎

######  Claim 4.

There exists a positive constant $\tau_{4}\;(\leq\tau_{0})$ such that if
$|z_{1}-z_{2}|\geq r$ and if either $|z_{1}|\leq R$ or $|z_{2}|\leq R$, then
$h_{t}(z_{1})\neq h_{t}(z_{2})$ for all $0\leq t\leq\tau_{4}$. Here, $r$ is
the constant chosen in Claim 1 and $R$ is the constant chosen in Claim 3.

###### Proof.

Suppose that there is no such $\tau_{4}$. Then, there exist a sequence
$\\{t_{n}\\}$ with $t_{n}\to 0$ as $n\to\infty$, and sequences $\\{z_{1n}\\}$,
$\\{z_{2n}\\}$ in $\mathbb{H}$ satisfying $|z_{1n}-z_{2n}|\geq r$ and either
$|z_{1n}|\leq R$ or $|z_{2n}|\leq R$, but
$h_{t_{n}}(z_{1n})=h_{t_{n}}(z_{2n})$ for all $n$. We may assume that
$\\{z_{1n}\\}$ and $\\{z_{2n}\\}$ converge to some points $z_{1}$ and $z_{2}$
in $\mathbb{H}\cup i\hat{\mathbb{R}}$ respectively by passing to subsequences.

By Claim  ‣ 4, $h_{t_{n}}(z_{1n})\to h(z_{1})$ and $h_{t_{n}}(z_{2n})\to
h(z_{2})$ as $n\to\infty$. Since $h_{t_{n}}(z_{1n})=h_{t_{n}}(z_{2n})$, we
have $h(z_{1})=h(z_{2})$. However, both the conditions that
$|z_{1n}-z_{2n}|\geq r$ and that either $|z_{1n}|\leq R$ or $|z_{2n}|\leq R$
imply that $z_{1}\neq z_{2}$. This contradicts the fact that $h$ is injective
in $\mathbb{H}\cup i\hat{\mathbb{R}}$. ∎

We will finish the proof. Set
$\tau:=\min\\{\tau_{1},\tau_{2},\tau_{3},\tau_{4}\\}>0$. Let $r$ be the
constant chosen in Claim 1, and $R$ the constant chosen in Claim 3. We show
that $h_{t}$ is univalent in $\mathbb{H}$ for $0\leq t\leq\tau$. For any
distinct points $z_{1}$ and $z_{2}$ in $\mathbb{H}$, one of the following
three cases occurs:

  * •

If $|z_{1}-z_{2}|<r$, then $h_{t}(z_{1})\neq h_{t}(z_{2})$ by Claims 1 and 2.

  * •

If $|z_{1}-z_{2}|\geq r$ and if both $|z_{1}|>R$ and $|z_{2}|>R$, then
$h_{t}(z_{1})\neq h_{t}(z_{2})$ by Claim 3.

  * •

If $|z_{1}-z_{2}|\geq r$ and if either $|z_{1}|\leq R$ or $|z_{2}|\leq R$,
then $h_{t}(z_{1})\neq h_{t}(z_{2})$ by Claim 4.

In any case, we have $h_{t}(z_{1})\neq h_{t}(z_{2})$ and hence $h_{t}$ is
injective in $\mathbb{H}$. This completes the proof of Theorem 4.3. ∎

Next, we prove the univalence of $h_{t}$ under assumption $\mathbf{(B)}$.

######  Theorem 4.5.

Let $h$ be univalent and analytic in $\mathbb{H}$ with
$\lim_{z\to\infty}h(z)=\infty$ and let $h(\mathbb{H})$ be a Jordan domain. Let
$h$ satisfy (1.5). Suppose $\mathbf{(B)}$: $h$ is locally quasiconformally
extendible to a neighborhood of $\infty$. Then, there is a positive constant
$\tau\leq\tau_{0}$ such that the analytic function $h_{t}$ on $\mathbb{H}$
defined by (4.2) is univalent for all $0\leq t\leq\tau$.

###### Proof.

In the proof of Theorem 4.3, assumption $\mathbf{(A)}$ is only used in Claim
3; the other claims are applicable also to the proof of Theorem 4.5. Hence, it
is enough to show the following parallel result, Claim 5, to Claim 3 and reset
$\tau:=\min\\{\tau_{1},\tau_{2},\tau_{4},\tau_{5}\\}$ and $R$, where
$\tau_{5}$ and $R$ are determined there. ∎

######  Claim 5.

Suppose that assumption $\mathbf{(B)}$ holds. Namely, $h$ has a quasiconformal
extension to $D_{M}:=\\{z\in\mathbb{C}:|z|>M\\}$ for some $M>0$. Then, there
exist positive constant $\tau_{5}\;(\leq\tau_{0})$ and $R>M$ such that if
$z_{1},z_{2}\in\Omega_{R}$ with $|z_{1}-z_{2}|\geq r$, then $h_{t}(z_{1})\neq
h_{t}(z_{2})$ for all $0\leq t\leq\tau_{5}$, where $r$ is the constant chosen
in Claim 1.

The key concept we use to prove Claim 5 is that of quasisymmetry which was
introduced by Beurling and Ahlfors [5] on the real line and formulated for
general metric spaces by Tukia and Väisälä [37]. For our purpose we only
consider it for the complex plane. We refer to [2, Definition 3.2.1].

######  Definition.

Let $D\subset\mathbb{C}$ be an open subset and $f:D\to\mathbb{C}$ an
orientation-preserving mapping. Let $\eta:[0,\,+\infty)\to[0,\,+\infty)$ be an
increasing homeomorphism with $\lim_{t\to+\infty}\eta(t)=+\infty$. We say that
$f$ is $\eta$-quasisymmetric if for each triple $z_{0}$, $z_{1}$, $z_{2}\in D$
we have

$\frac{|f(z_{0})-f(z_{1})|}{|f(z_{0})-f(z_{2})|}\leq\eta\left(\frac{|z_{0}-z_{1}|}{|z_{0}-z_{2}|}\right).$

If $f:\mathbb{C}\to\mathbb{C}$ is a $k$-quasiconformal homeomorphism of
$\mathbb{C}$, then $f$ is $\eta$-quasisymmetric where $\eta$ depends only on
$k$ (see [2, Theorem 3.5.3]); conversely, if $f:D\to\mathbb{C}$ is an
$\eta$-quasisymmetric mapping on a domain $D$ then $f$ is quasiconformal (see
[2, Theorem 3.4.1]).

###### Proof of Claim 5.

Suppose that $h$ has a quasiconformal extension to $D_{M}$, which we still
denote by $h$. Let $R=M+1$. By an extension theorem (see [28, Theorem
II.1.8]), there exists a quasiconformal homeomorphism $F$ of the whole plane
$\mathbb{C}$ that coincides with $h$ in $D_{R}$. In particular, it coincides
with $h$ in $\Omega_{R}$. The inverse $F^{-1}$ of $F$ is also a quasiconformal
homeomorphism of $\mathbb{C}$, and then it is a quasisymmetric homeomorphism
of $\mathbb{C}$. Thus, there exists an increasing homeomorphism
$\eta:[0,+\infty)\to[0,+\infty)$ such that

$\frac{|F^{-1}(w_{0})-F^{-1}(w_{1})|}{|F^{-1}(w_{0})-F^{-1}(w_{2})|}\leq\eta\left(\frac{|w_{0}-w_{1}|}{|w_{0}-w_{2}|}\right)$

for any $w_{0},w_{1},w_{2}\in\mathbb{C}$. We take the inverse
$\lambda:=\eta^{-1}$.

For any $z_{1},z_{2}\in\Omega_{R}$ with $|z_{1}-z_{2}|\geq r$ and any $t>0$,
we have

$\begin{split}\frac{|h(z_{1}+t)-h(z_{2}+t)|}{|h(z_{1})-h(z_{1}+t)|}&=\frac{|F(z_{1}+t)-F(z_{2}+t)|}{|F(z_{1})-F(z_{1}+t)|}\\\
&\geq\lambda\left(\frac{|(z_{1}+t)-(z_{2}+t)|}{|z_{1}-(z_{1}+t)|}\right)=\lambda\left(\frac{|z_{1}-z_{2}|}{t}\right)\geq\lambda\left(\frac{r}{t}\right).\end{split}$
(4.19)

We may assume that $|h^{\prime}(z_{1}+t)|\geq|h^{\prime}(z_{2}+t)|$ by
exchanging the roles of $z_{1}$ and $z_{2}$. Then, by using (4.4), (4.16), and
(4.19), we obtain that

$\begin{split}|h_{t}(z_{1})-h_{t}(z_{2})|&=\left|h(z_{1}+t)-h(z_{2}+t)-\frac{2th^{\prime}(z_{1}+t)}{1+tPh(z_{1}+t)}+\frac{2th^{\prime}(z_{2}+t)}{1+tPh(z_{2}+t)}\right|\\\
&\geq|h(z_{1}+t)-h(z_{2}+t)|-4t|h^{\prime}(z_{1}+t)|-4t|h^{\prime}(z_{2}+t)|\\\
&\geq|h(z_{1})-h(z_{1}+t)|\lambda\left(\frac{r}{t}\right)-8t|h^{\prime}(z_{1}+t)|\\\
&\geq\frac{t}{4}|h^{\prime}(z_{1}+t)|\lambda\left(\frac{r}{t}\right)-8t|h^{\prime}(z_{1}+t)|=\frac{t}{4}|h^{\prime}(z_{1}+t)|\left(\lambda\left(\frac{r}{t}\right)-32\right)\end{split}$

for all $0<t\leq\tau_{0}$. It can be seen from the monotonicity of $\lambda$
that there exists a positive constant $\tau_{4}\;(\leq\tau_{0})$ such that
$\lambda(r/t)>32$ for all $0<t\leq\tau_{4}$. This shows that $h_{t}(z_{1})\neq
h_{t}(z_{2})$ for all $0\leq t\leq\tau_{4}$. ∎

We have proved Theorems 4.3 and 4.5. Then, Theorem 1.2 follows from these
theorems combined with Theorem 4.1 and Lemma 4.2.

###### Proof of Theorem 1.2..

By Theorems 4.3 and 4.5, we see that the analytic function $h_{t}$ defined by
(4.2) is univalent in $\mathbb{H}$ for each $0\leq t\leq\tau$. Then by Lemma
4.2, we obtain that $(h_{t})$ is a chordal Loewner chain over the interval
$[0,\tau)$ with the associated Herglotz function $p$ satisfying (4.1) (see
[15, Theorem 4.1]). From Theorem 4.1, the assertion of Theorem 1.2 follows.
Indeed, the extension $\hat{h}$ is defined by

$\hat{h}(z)=h_{-{\rm Re}z}(i{\rm Im}z),\qquad-\tau<{\rm Re}\,z<0,$

for the chordal Loewner chain $(h_{t})$ over the interval $[0,\tau)$ given by
(4.2). This yields the explicit formula of this extension in (1.6). From this,
its complex dilatation can be computed directly as in (1.7). ∎

##  5\. Quasiconformal extensions by means of pre-Schwarzian derivative

We have the following parallel result to Theorems 1.2 by replacing the
Schwarzian derivative $Sf$ of $f$ with the pre-Schwarzian derivative $Pf$ of
$f$.

######  Theorem 5.1.

Let $f$ be univalent and analytic in $\mathbb{H}$ with
$\lim_{z\to\infty}f(z)=\infty$ such that $f(\mathbb{H})$ is a Jordan domain.
Let $f$ satisfy the condition

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)\left|Pf(z)\right|=0.$

Suppose further that either $\mathbf{(A)}$ or $\mathbf{(B)}$ holds. Then, the
function

$\hat{f}(z)=\begin{cases}f(z),&\text{if}\;\;{\rm Re}\,z\geq 0,\\\
f(z^{*})+(2{\rm
Re}z)f^{\prime}(z^{*}),\quad(z^{*}:=-\bar{z})&\text{if}\;\;-\tau<{\rm
Re}\,z<0\end{cases}$

defines a quasiconformal extension of $f$ over $i\mathbb{R}$ to
$\mathbb{H}^{*}_{(0,\tau)}:=\\{z\in\mathbb{C}:-\tau<{\rm Re}\,z<0\\}$ for some
$\tau>0$ such that its complex dilatation $\mu$ on $\mathbb{H}^{*}_{(0,\tau)}$
has the form

$\mu(z)=-(2{\rm Re}z)Pf(z^{*}).$

The proof of Theorem 5.1 is completely similar to that of Theorem 1.2. We only
make mention of the difference. Instead of considering the family $(h_{t})$
defined by (4.2), we consider a simpler one

$f_{t}(z):=f(z+t)-2tf^{\prime}(z+t),\qquad z\in\mathbb{H},\quad 0\leq
t\leq\tau_{0}$

for some $\tau_{0}>0$. It is associated with the Herglotz function

$p(z,t)=\frac{1+2tPf(z+t)}{1-2tPf(z+t)}$

in comparison with (4.8). This satisfies

$\left|\frac{p(z,t)-1}{p(z,t)+1}\right|=2t\left|Pf(z+t)\right|\leq k<1$

for all $z\in\mathbb{H}$ and all $0\leq t\leq\tau_{0}$ in comparison with
(4.9). Then, the corresponding statement to Lemma 4.2 is obtained. To prove
that there is some positive constant $\tau\;(\leq\tau_{0})$ such that $f_{t}$
is univalent for any $0\leq t\leq\tau$, we repeat the arguments for Theorems
4.3 and 4.5. Because of dealing with the simpler family $(f_{t})$, the
argument becomes a bit simpler. We omit the details here.

##  6\. Application to VMO-Teichmüller space (proof of Theorem 1.4)

Before proceeding to the proof of Theorem 1.4, let us recall some basic
definitions on Carleson measures and BMO functions (see [20, Chapter 6]).

We say that a positive measure $\lambda$ on $\mathbb{H}$ is a Carleson measure
if

$\|\lambda\|_{c}:=\sup_{I\subset i\mathbb{R}}\frac{\lambda((0,\,|I|)\times
I)}{|I|}<\infty,$

where the supremum is taken over all bounded intervals $I$ in $i\mathbb{R}$. A
Carleson measure $\lambda$ is called a vanishing Carleson measure if

$\lim_{|I|\to 0}\frac{\lambda((0,\,|I|)\times I)}{|I|}=0.$

We denote by ${\rm CM}(\mathbb{H})$ and ${\rm CM}_{0}(\mathbb{H})$ the set of
all Carleson measures and vanishing Carleson measures on $\mathbb{H}$,
respectively. ${\rm CM}(\mathbb{H}^{*})$ and ${\rm CM}_{0}(\mathbb{H}^{*})$
can be defined similarly.

A locally integrable complex-valued function $u$ on $i\mathbb{R}$ is of BMO
(denoted by $u\in{\rm BMO}(i\mathbb{R})$) if

$\|u\|_{*}:=\sup_{I\subset
i\mathbb{R}}\frac{1}{|I|}\int_{I}|u(z)-u_{I}||dz|<\infty,$

where the supremum is taken over all bounded intervals $I$ on $i\mathbb{R}$
and $u_{I}$ denotes the integral mean of $u$ over $I$. Moreover, $u$ is of VMO
(denoted by $u\in{\rm VMO}(i\mathbb{R})$) if in addition

$\lim_{|I|\to 0}\frac{1}{|I|}\int_{I}|u(z)-u_{I}||dz|=0.$

Let $\rm BMOA(\mathbb{H})$ denote the space of all analytic functions $\phi$
on $\mathbb{H}$ that are Poisson integrals of BMO functions on $i\mathbb{R}$,
and let $\rm VMOA(\mathbb{H})$ denote the subspace of $\rm BMOA(\mathbb{H})$
whose elements have boundary values in ${\rm VMO}(i\mathbb{R})$. By using the
similar arguments to the case of the unit disk (see [20, p.233]), we can
conclude that an analytic function $\phi$ on $\mathbb{H}$ belongs to $\rm
BMOA(\mathbb{H})$ if and only if $\phi$ induces a Carleson measure

$(2{\rm Re}z)|\phi^{\prime}(z)|^{2}dxdy\in{\rm CM}(\mathbb{H}),$

and moreover, $\phi$ belongs to $\rm VMOA(\mathbb{H})$ if and only if

$(2{\rm Re}z)|\phi^{\prime}(z)|^{2}dxdy\in{\rm CM}_{0}(\mathbb{H}).$

We need the following result from [35, Proposition 7.4]. This is conceptually
similar to Proposition 4.4.

######  Proposition 6.1.

Let $\psi$ be an analytic function on $\mathbb{H}$ such that
$\lim_{x\to+\infty}\psi(x+iy)=0$ uniformly for $y\in\mathbb{R}$. For
$\alpha>0$ set

$\lambda_{1}:=|\psi(z)|^{2}x^{\alpha}dxdy$

and

$\lambda_{2}:=|\psi^{\prime}(z)|^{2}x^{\alpha+2}dxdy$

for $z=x+iy\in\mathbb{H}$. Then, $\lambda_{1}\in{\rm CM}(\mathbb{H})$ if and
only if $\lambda_{2}\in{\rm CM}(\mathbb{H})$, and
$\|\lambda_{1}\|_{c}\asymp\|\lambda_{2}\|_{c}$ with comparison constants
depending only on $\alpha$. Moreover, $\lambda_{1}\in{\rm CM}_{0}(\mathbb{H})$
if and only if $\lambda_{2}\in{\rm CM}_{0}(\mathbb{H})$.

###### Proof of Theorem 1.4.

Since $h$ admits a quasiconformal extension to $\overline{\mathbb{C}}$ with
$h(\infty)=\infty$, we have the well-known inequalities (see [9] for more
information):

$(2{\rm Re}z)|Ph(z)|<6;\quad(2{\rm Re}z)^{2}|Sh(z)|<6$ (6.1)

for all $z\in\mathbb{H}$. These in particular yield that $Ph(z)\to 0$ and
$Sh(z)\to 0$ uniformly for ${\rm Im}\,z\in\mathbb{R}$ as ${\rm
Re}\,z\to+\infty$. Since $\log h^{\prime}\in{\rm VMOA}(\mathbb{H})$, we obtain
by [35, Lemma 7.1] that

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)\left|Ph(z)\right|=0,$

and then by Theorem 3.5 that

$\lim_{{\rm Re}z\to 0^{+}}(2{\rm Re}z)^{2}\left|Sh(z)\right|=0.$

Thus, $h$ satisfies the condition of Corollary 1.3. In addition, the Loewner
chain $(h_{t})$ over some interval $[0,\tau)$ produced by $h$ in the form of
(4.2) satisfies the condition of Theorem 4.1.

It follows from the equality $Sh=(Ph)^{\prime}-(Ph)^{2}/2$ and (6.1) that

$\begin{split}(2{\rm Re}z)^{3}|Sh(z)|^{2}&\leq 2(2{\rm
Re}z)^{3}|(Ph(z))^{\prime}|^{2}+1/2\left((2{\rm Re}z)|Ph(z)|\right)^{2}(2{\rm
Re}z)|Ph(z)|^{2}\\\ &\leq 2(2{\rm Re}z)^{3}|(Ph(z))^{\prime}|^{2}+18(2{\rm
Re}z)|Ph(z)|^{2}.\end{split}$ (6.2)

Since $\log h^{\prime}\in{\rm VMOA}(\mathbb{H})$, or equivalently,

$(2{\rm Re}z)|Ph(z)|^{2}dxdy\in{\rm CM}_{0}(\mathbb{H}),$ (6.3)

we conclude by Proposition 6.1 and (6.2) that

$(2{\rm Re}z)^{3}|Sh(z)|^{2}dxdy\in{\rm CM}_{0}(\mathbb{H}).$ (6.4)

We choose any $0<t<\tau$ and fix it. It follows from (4.7) that

$Ph_{t}(z)=\left(\log
h_{t}^{\prime}(z)\right)^{\prime}=Ph(z+t)+\frac{2t^{2}(Sh(z+t))^{\prime}}{1+2t^{2}Sh(z+t)}-\frac{2t(Ph(z+t))^{\prime}}{1+tPh(z+t)}.$

By means of (4.4) and (4.5), $|1+2t^{2}Sh(z+t)|$ and $|1+tPh(z+t)|$ are
bounded from below by a positive constant. Then, this yields that

$\begin{split}(2{\rm Re}z)|Ph_{t}(z)|^{2}\leq&C((2{\rm
Re}(z+t))|Ph(z+t)|^{2}\\\ &\quad+(2{\rm
Re}(z+t))^{5}|(Sh(z+t))^{\prime}|^{2}+(2{\rm
Re}(z+t))^{3}|(Ph(z+t))^{\prime}|^{2}\,)\end{split}$

for some positive constant $C$. Making use of Proposition 6.1, we also
conclude by (6.3) and (6.4) that

$(2{\rm Re}z)|Ph_{t}(z)|^{2}dxdy\in{\rm CM}(\mathbb{H}).$ (6.5)

This is equivalent to saying that $\log(h_{t})^{\prime}$ belongs to $\rm
BMOA(\mathbb{H})$.

Since $h$ admits quasiconformal extensions to $\mathbb{C}$ with
$\lim_{z\to\infty}h(z)=\infty$ and so does $\varphi_{0,t}$ by Theorem 4.1
(iii), we see that $h_{t}=h\circ\varphi_{0,t}^{-1}$ can be quasiconformally
extended to $\overline{\mathbb{C}}$ with $h_{t}(\infty)=\infty$. In fact, in
virtue of property (6.5), this further implies that $h_{t}$ admits such a
particular quasiconformal extension $\widehat{h_{t}}$ that its complex
dilatation $\mu_{t}$ on $\mathbb{H}^{*}$ induces a Carleson measure

$|\mu_{t}(z)|^{2}/(-2{\rm Re}z)dxdy\in{\rm CM}(\mathbb{H}^{*})$ (6.6)

(see [35, Theorem 7.2]). For example, it is known that the variant of the
Beurling–Ahlfors extension by heat kernel (see [19, Theorem 4.2], [39, Theorem
3.4]) realizes such an extension.

Mediated by the relation $\hat{h}(-t+iy)=h_{t}(iy)$, a map $\tilde{h}$ on
$\mathbb{H}^{*}$ is composed by

$\tilde{h}(z):=\begin{cases}h(z),&\text{if}\;\;{\rm Re}z\geq 0,\\\ h_{-{\rm
Re}z}(i{\rm Im}z)=h(z^{*})+\frac{(2{\rm Re}z)h^{\prime}(z^{*})}{1-({\rm
Re}z)Ph(z^{*})},&\text{if}\;\;-t\leq{\rm Re}z<0,\\\
\widehat{h_{t}}(z+t),&\text{if}\;\;{\rm Re}z<-t.\end{cases}$

This is well-defined, and yields a quasiconformal extension of $h$ to
$\mathbb{C}$ whose complex dilatation $\tilde{\mu}$ on $\mathbb{H}^{*}$ is
given by

$\tilde{\mu}(z)=\begin{cases}-\frac{1}{2}(2{\rm
Re}z)^{2}Sh(z^{*}),&\text{if}\;\;-t\leq{\rm Re}z<0,\\\
\mu_{t}(z+t),&\text{if}\;\;{\rm Re}z<-t.\end{cases}$

We will show that $\tilde{\mu}$ induces a vanishing Carleson measure on
$\mathbb{H}^{*}$ as in (1.9). For an interval $I\subset i\mathbb{R}$ with
$|I|\leq t$, we see from (6.4) that

$\frac{1}{|I|}\iint_{(-|I|,\,0)\times I}\frac{|\tilde{\mu}(z)|^{2}}{(-2{\rm
Re}z)}dxdy=\frac{1}{4|I|}\iint_{(0,\,|I|)\times I}(2{\rm
Re}z)^{3}|Sh(z)|^{2}dxdy$

is bounded uniformly with respect to $I$, and tends to $0$ as $|I|\to 0$. For
an interval $I\subset i\mathbb{R}$ with $|I|>t$, we have

$\begin{split}\frac{1}{|I|}\iint_{(-|I|,\,0)\times
I}\frac{|\tilde{\mu}(z)|^{2}}{-2{\rm
Re}z}dxdy=&\frac{1}{4|I|}\iint_{(-t,0)\times I}(2{\rm
Re}z)^{2}|Sh(z)|^{2}dxdy\\\ &+\frac{1}{|I|}\iint_{(-|I|,-t]\times
I}\frac{|\mu_{t}(z+t)|^{2}}{(-2{\rm Re}z)}dxdy.\end{split}$ (6.7)

By (6.4) and (6.6), it is not difficult to see that this is bounded uniformly
with respect to $I$. Indeed, for the estimate of the first term of the right
side of (6.7), we divide $[-t,0)\times I$ into $n$ congruent rectangles
$(-t,0)\times I_{i}$ $(i=1,\ldots,n)$ so that $nt\leq|I|<(n+1)t$. For the
estimate of the second term, we replace $(-|I|,-t]\times I$ with a larger
square $(-|I|-t,-t]\times I$ and translate it by $t$ along $x$-axis.
Consequently, $|\tilde{\mu}(z)|^{2}/(-2{\rm Re}z)dxdy$ is a vanishing Carleson
measure on $\mathbb{H}^{*}$. ∎

######  Remark.

As mentioned in the introduction, this result completes the characterization
of the elements of the VMO Teichmüller space on the upper half-plane developed
by Shen [35]. In addition, we can show that the image of the space of Beltrami
coefficients inducing vanishing Carleson measures on $\mathbb{H}^{*}$ by the
Schwarzian derivative map coincides with the intersection of the Bers
embedding of the universal Teichmüller space $T$ and the space of the
Schwarzian derivatives inducing vanishing Carleson measures on $\mathbb{H}$.
This answers a question in [35, Remark 5.2].

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