††footnotetext:  File: main.tex, printed: 2024-9-2, 23.01

# On the subinvariance of uniform domains in metric spaces

Yaxiang Li  Yaxiang Li, College of Science, Central South University of
Forestry and Technology, Changsha, Hunan 410004, People’s Republic of China
<EMAIL_ADDRESS>,  Manzi Huang  Manzi Huang, Department of Mathematics,
Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
<EMAIL_ADDRESS>,  Xiantao Wang∗ Xiantao Wang, Department of
Mathematics, Shantou University, Shantou, Guangdong 515063, People’s Republic
of China<EMAIL_ADDRESS>and  Qingshan Zhou  Qingshan Zhou, Department of
Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s
Republic of China<EMAIL_ADDRESS>
###### Abstract.

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that
$G\subset X$ and $G^{\prime}\subset Y$ are domains, and that $f:G\to
G^{\prime}$ is a homeomorphism. Our main result is the following subinvariance
property of the class of uniform domains: Suppose both $f$ and $f^{-1}$ are
weakly quasisymmetric mappings and $G^{\prime}$ is a quasiconvex domain. Then
the image $f(D)$ of every uniform subdomain $D$ in $G$ under $f$ is uniform.
The subinvariance of uniform domains with respect to freely quasiconformal
mappings or quasihyperbolic mappings is also studied with the additional
condition that both $G$ and $G^{\prime}$ are locally John domains.

###### Key words and phrases:

Subinvariance, uniform domain, weak quasisymmetry, free quasiconformality,
quasihyperbolic homemorphism, locally John domain.
∗ Corresponding author

###### 2000 Mathematics Subject Classification:

Primary: 30C65, 30F45; Secondary: 30C20

The research was partly supported by NNSF of China

##  1\. Introduction and main results

The quasihyperbolic metric (briefly, QH metric) was introduced by Gehring and
his students Palka and Osgood in the 1970’s [12, 13] in the setting of
Euclidean spaces ${\mathbb{R}}^{n}$ $(n\geq 2).$ Since its first appearance,
the quasihyperbolic metric has become an important tool in the geometric
function theory of Euclidean spaces, especially, in the study of
quasiconformal and quasisymmetric mappings. From late 1980’s onwards, Väisälä
has developed the theory of (dimension) free quasiconformal mappings (briefly,
free theory) in Banach spaces, which is based on properties of the
quasihyperbolic metric [36, 37, 38, 39, 41]. The main advantage of this
approach over generalizations based on the conformal modulus (see [16] and
references therein) is that it does not make use of volume integrals, thus
allowing the study of quasiconformality in infinite dimensional Banach spaces.

The class of quasisymmetric mappings on the real axis was first introduced by
Beurling and Ahlfors [4], who found a way to obtain a quasiconformal extension
of a quasisymmetric self-mapping of the real axis to a self-mapping of the
upper half-plane. This idea was later generalized by Tukia and Väisälä, who
studied quasisymmetric mappings between metric spaces [32]. In 1998, Heinonen
and Koskela [18] proved a remarkable result, showing that the concepts of
quasiconformality and quasisymmetry are quantitatively equivalent in a large
class of metric spaces, which includes Euclidean spaces. Also, Väisälä proved
the quantitative equivalence between free quasiconformality and quasisymmetry
of homeomorphisms between two Banach spaces. See [41, Theorem 7.15]. Against
this background, it is not surprising that the study of quasisymmetry in
metric spaces has recently attracted significant attention.

The main goal of this paper is to establish the subinvariance of uniform
domains in suitable metric spaces with respect to quaisymmetric mappings,
freely quasiconformal mappings and quasihyperbloic mappings. We start by
recalling some basic definitions. Through this paper, we always assume that
$X$ and $Y$ are metric spaces. We follow the notations and terminology of [17,
18, 25, 33, 41].

######  Definition 1.

A homeomorphism $f$ from $X$ to $Y$ is said to be

  1. (1)

quasiconformal if there is a constant $H<\infty$ such that

(1.1) $\limsup_{r\to 0}\frac{L_{f}(x,r)}{l_{f}(x,r)}\leq H$

for all $x\in X$;

  2. (2)

quasisymmetric if there is a constant $H<\infty$ such that

(1.2) $\frac{L_{f}(x,r)}{l_{f}(x,r)}\leq H$

for all $x\in X$ and all $r>0$,

where $L_{f}(x,r)=\sup_{|y-x|\leq r}\\{|f(y)-f(x)|\\}$ and
$l_{f}(x,r)=\inf_{|y-x|\geq r}\\{|f(y)-f(x)|\\}$.

Here and in what follows, we always use $|x-y|$ to denote the distance between
$x$ and $y$.

######  Definition 2.

A homeomorphism $f$ from $X$ to $Y$ is said to be

  1. (1)

$\eta$-quasisymmetric if there is a homeomorphism
$\eta:[0,\infty)\to[0,\infty)$ such that

$|x-a|\leq t|x-b|\;\;\mbox{implies}\;\;|f(x)-f(a)|\leq\eta(t)|f(x)-f(b)|$

for each $t\geq 0$ and for each triple $x,$ $a$, $b$ of points in $X$;

  2. (2)

weakly $H$-quasisymmetric if

$|x-a|\leq|x-b|\;\;\mbox{ implies}\;\;|f(x)-f(a)|\leq H|f(x)-f(b)|$

for each triple $x$, $a$, $b$ of points in $X$.

######  Remark 1.

The following observations follow immediately from Definitions 1 and 2.

  1. (1)

The quasisymmetry implies the quasiconformality;

  2. (2)

A homeomorphism $f$ from $X$ to $Y$ is quasisymmetric with coefficient $H$
defined by Definition 1(2) if and only if it is weakly $H$-quasisymmetric;

  3. (3)

The $\eta$-quasisymmetry implies the weak $H$-quasisymmetry with $H=\eta(1)$.
Obviously, $\eta(1)\geq 1$. In general, the converse is not true (cf. [41,
Theorem $8.5$]). See also [22] for the related discussions.

The definition of free quasiconformality is as follows.

######  Definition 3.

Let $G\varsubsetneq X$ and $G^{\prime}\varsubsetneq Y$ be two domains (open
and connected), and let $\varphi:[0,\infty)\to[0,\infty)$ be a homeomorphism
with $\varphi(t)\geq t$. We say that a homeomorphism $f:G\to G^{\prime}$ is

  1. (1)

$\varphi$-semisolid  if

$k_{G^{\prime}}(f(x),f(y))\leq\varphi(k_{G}(x,y))$

for all $x$, $y\in G$;

  2. (2)

$\varphi$-solid if both $f$ and ${\it f^{-1}}$ are $\varphi$-semisolid;

  3. (3)

freely $\varphi$-quasiconformal ($\varphi$-FQC in brief) or fully
$\varphi$-solid if $f$ is $\varphi$-solid in every subdomain of $G$,

where $k_{G}(x,y)$ denotes the quasihyperbolic distance of $x$ and $y$ in $G$.
See Section 2 for the precise definitions of $k_{G}(x,y)$ and other notations
and concepts in the rest of this section.

It follows from [9, Remark, p. 121] and [34, Theorem 5.6] that uniform domains
are subinvariant with respect to quasiconformal mappings in ${\mathbb{R}}^{n}$
($n\geq 2$). By this, we mean that if $f:\;G\to G^{\prime}$ is a
$K$-quasiconformal mapping, where $G$ and $G^{\prime}$ are domains in
${\mathbb{R}}^{n}$, and if $G^{\prime}$ is $c$-uniform, then $D^{\prime}=f(D)$
is $c^{\prime}$-uniform for every $c$-uniform subdomain $D\subset G$, where
$c^{\prime}=c^{\prime}(c,K,n)$ which means that the constant $c^{\prime}$
depends only on the coefficient $c$ of the uniformity of $D$, the coefficient
$K$ of quasiconformality of $f$ and the dimension $n$ of the Euclidean space
${\mathbb{R}}^{n}$. In the free theory, Väisälä also studied this property of
uniform domains in Banach spaces and proved the following result.

Theorem A. $($[40, Theorem 2.44]$)$ Suppose that $G\subset E$ and
$G^{\prime}\subset E^{\prime}$, where $E$ and $E^{\prime}$ are Banach spaces
with dimension at least $2$, that the domains $G^{\prime}$ and $D\subset G$
are $c$-uniform, and that $f:\;G\to G^{\prime}$ is $M$-quasihyperbolic. Then
$f(D)$ is $c^{\prime}$-uniform with $c^{\prime}=c^{\prime}(c,M)$.

In 2012, Huang, Vuorinen and Wang proved the subinvariance property of uniform
domains is also true with respect to freely quasiconformal mappings ([24,
Theorem 1]). See [7, 11, 34, 41, 43] for similar discussions in this line.

Our work is motivated by the above ideas which we will extend to the context
of weakly quasisymmetric mappings, freely quasiconformal mappings and
quasihyperbolic mappings in metric spaces. Our first result is as follows.

######  Theorem 1.

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that
$G\varsubsetneq X$ is a domain, $G^{\prime}\varsubsetneq Y$ is a quasiconvex
domain, and that both $f:G\to G^{\prime}$ and $f^{-1}:G^{\prime}\to G$ are
weakly quasisymmetric mappings. For each subdomain $D$ of $G$, if $D$ is
uniform, then $D^{\prime}=f(D)$ is uniform, where the coefficient of
uniformity of $D^{\prime}$ depends only on the given data of $X$, $Y$, $G$,
$G^{\prime}$, $D$, $f$ and $f^{-1}$.

Here and in what follows, the phrase “the given data of $X$, $Y$, $G$,
$G^{\prime}$, $D$, $f$ and $f^{-1}$” means the data which depends on the given
constants which are the coefficients of quasiconvexity of $X$, $Y$ and
$G^{\prime}$, the coefficient of uniformity of $G$ and the coefficients of
weak quasisymmetry of $f$ and $f^{-1}$.

######  Remark 2.

It is worth to mention that in Theorem 1, the domain $G^{\prime}$ is not
required to be “uniform”, and only to be “quasiconvex” (From the definitions
in Section 2, we easily see that uniformity implies quasiconvexity). Moreover,
we see from the example constructed in the paragraph next to Theorem 1.1 in
[43] that the assumption “quasiconvexity” in Theorem 1 is necessary.

As we have indicated in the second paragraph that quasisymmetry and
quasiconformality are quantitatively equivalent for homeomorphisms between
${\mathbb{R}}^{n}$. Also, it follows from [12] and [32] that quasiconformality
and free quasiconformality are quantitatively equivalent for homeomorphisms
between domains in ${\mathbb{R}}^{n}$. These facts together with [24, Theorem
1] prompt us to conjecture that whether Theorem 1 still holds if we replace
the assumption “weakly quasisymmetric mappings” (resp. “$G^{\prime}$ being
quasiconvex”) by the one “freely quasiconformal mappings” (resp. “$G^{\prime}$
being uniform”). Under the extra assumption “$G$ and $G^{\prime}$ being
locally John”, we have the following partial answer to this problem.

######  Theorem 2.

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that
$G\varsubsetneq X$ is a non-point-cut and locally John domain, and that
$G^{\prime}\varsubsetneq Y$ is an uniform and locally John domain. If $f:$
$G\to G^{\prime}$ is a freely quasiconformal mapping, then for each uniform
subdomain $D$ in $G$, its image $D^{\prime}=f(D)$ must be uniform, where the
coefficient of uniformity of $D^{\prime}$ depends only on the given data of
$X$, $Y$, $G$, $G^{\prime}$, $D$ and $f$.

Here $G$ is said to be non-point-cut if for any $x\in G$, the set
$G\backslash\\{x\\}$ is a subdomain of $G$.

######  Remark 3.

When $X=Y={\mathbb{R}}^{n}$, Theorem 2 is the subinvariance of uniform domains
with respect to quasiconformal mappings in ${\mathbb{R}}^{n}$ ([9, Remark,
P.121] and [34, Theorem 5.6]). Also, when $X$ and $Y$ are Banach spaces with
dimension at least $2$, Theorem 2 coincides with [24, Theorem 1].

Also, we get the following subinvariance of uniform domains with respect to
quasihyperbolic mappings.

######  Theorem 3.

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that
$G\varsubsetneq X$ is a non-point-cut and locally John domain, and that
$G^{\prime}\varsubsetneq Y$ is an uniform and locally John domain. If $f:$
$G\to G^{\prime}$ is a quasihyperbolic mapping, then for each uniform
subdomain $D$ in $G$, its image $D^{\prime}=f(D)$ must be uniform, where the
coefficient of uniformity of $D^{\prime}$ depends only on the given data of
$X$, $Y$, $G$, $G^{\prime}$, $D$ and $f$.

######  Remark 4.

When $X$ and $Y$ are Banach spaces with dimension at least $2$, Theorem 3
coincides with Theorem , i.e., Theorem 2.44 in [40].

We also conjecture that whether there is the subinvariance of John domains in
suitable metric spaces with respect to weakly quasisymmetric mappings, freely
quasiconformal mappings etc. See [15, 21] etc for the related discussions in
${\mathbb{R}}^{n}$.

The rest of this paper is organized as follows. In Section 2, we recall some
definitions and preliminary results. In Section 3, the proof of Theorem 1 is
given. Section 4 is devoted to the proof of Theorem 2, and the proof of
Theorem 3 is presented in Section 5.

##  2\. Preliminaries

In this section, we give the necessary definitions and auxiliary results,
which will be used in the proofs of our main results.

Throughout this paper, balls and spheres in metric spaces $X$ are written as

$\mathbb{B}(a,r)=\\{x\in X:\,|x-a|<r\\},\;\;\mathbb{S}(a,r)=\\{x\in
X:\,|x-a|=r\\}$

and

$\mathbb{\overline{B}}(a,r)=\mathbb{B}(a,r)\cup\mathbb{S}(a,r)=\\{x\in
X:\,|x-a|\leq r\\}.$

For convenience, given domains $G\subset X,$ $G^{\prime}\subset Y$, a map
$f:G\to G^{\prime}$ and points $x$, $y$, $z$, $\ldots$ in $G$, we always
denote by $x^{\prime}$, $y^{\prime}$, $z^{\prime}$, $\ldots$ the images in
$G^{\prime}$ of $x$, $y$, $z$, $\ldots$ under $f$, respectively. Also, we
assume that $\gamma$ denotes an arc in $G$ and $\gamma^{\prime}$ the image in
$G^{\prime}$ of $\gamma$ under $f$.

###  2.1. Quasihyperbolic metric, solid arcs and short arcs

In this subsection, we start with the definition of quasihyperbolic metric. If
$X$ is a connected metric space and $G\varsubsetneq X$ is a non-empty open
set, then it follows from [25, Remark 2.2] that the boundary of $G$ satisfies
$\partial G\not=\emptyset$. Suppose $\gamma\subset G$ denotes a rectifiable
arc or a path, its quasihyperbolic length is the number:

$\ell_{k_{G}}(\gamma)=\int_{\gamma}\frac{|dz|}{\delta_{G}(z)},$

where $\delta_{G}(z)$ denotes the distance from $z$ to $\partial G$.

For each pair of points $x$, $y$ in $G$, the quasihyperbolic distance
$k_{G}(x,y)$ between $x$ and $y$ is defined in the following way:

$k_{G}(x,y)=\inf\ell_{k_{G}}(\gamma),$

where the infimum is taken over all rectifiable arcs $\gamma$ joining $x$ to
$y$ in $G$.

Suppose $X$ is quasiconvex and $G\subsetneq X$. If $\gamma$ is a rectifiable
curve in $G$ connecting $x$ and $y$, then (see, e.g., the proof of Theorem
$2.7$ in [25])

$\ell_{k_{G}}(\gamma)\geq\log\Big{(}1+\frac{\ell(\gamma)}{\min\\{\delta_{G}(x),\delta_{G}(y)\\}}\Big{)}$

and thus,

$k_{G}(x,y)\geq\log\Big{(}1+\frac{|x-y|}{\min\\{\delta_{G}(x),\delta_{G}(y)\\}}\Big{)}.$

Gehring and Palka [13] introduced the quasihyperbolic metric of a domain in
${\mathbb{R}}^{n}$. For the basic properties of this metric we refer to [12].
Recall that a curve $\gamma$ from $x$ to $y$ is a quasihyperbolic geodesic if
$\ell_{k_{G}}(\gamma)=k_{G}(x,y)$. Each subcurve of a quasihyperbolic geodesic
is obviously a quasihyperbolic geodesic. It is known that a quasihyperbolic
geodesic between any two points in a Banach space $X$ exists if the dimension
of $X$ is finite, see [12, Lemma 1]. This is not true in arbitrary metric
spaces (cf. [36, Example 2.9]).

Let us recall a result which is useful for the discussions later on.

Lemma B. ([23, Lemma 2.4]) Let $X$ be a $c$-quasiconvex metric space and let
$G\subsetneq X$ be a domain. Suppose that $x$, $y\in G$ and either
$|x-y|\leq\frac{1}{3c}\delta_{G}(x)$ or $k_{G}(x,y)\leq 1$. Then

(2.1)
$\frac{1}{2}\frac{|x-y|}{\delta_{G}(x)}<k_{G}(x,y)<3c\frac{|x-y|}{\delta_{G}(x)}.$

Here, we say that $X$ is c-quasiconvex $(c\geq 1)$ if each pair of points $x$,
$y\in X$ can be joined by an arc $\gamma$ with length ${\ell}(\gamma)\leq
c|x-y|$.

######  Definition 4.

Suppose $\gamma$ is an arc in a domain $G\varsubsetneq X$ and $X$ is a
rectifiably connected metric space. The arc may be closed, open or half open.
Let $\overline{x}=(x_{0},$ $\ldots,$ $x_{n})$, $n\geq 1$, be a finite sequence
of successive points of $\gamma$. For $h\geq 0$, we say that $\overline{x}$ is
$h$-coarse if $k_{G}(x_{j-1},x_{j})\geq h$ for all $1\leq j\leq n$. Let
$\Phi_{k}(\gamma,h)$ denote the family of all $h$-coarse sequences of
$\gamma$. Set

$s_{k}(\overline{x})=\sum^{n}_{j=1}k_{G}(x_{j-1},x_{j})$

and

$\ell_{k_{G}}(\gamma,h)=\sup\\{s_{k}(\overline{x}):\overline{x}\in\Phi_{k}(\gamma,h)\\}$

with the agreement that $\ell_{k_{G}}(\gamma,h)=0$ if
$\Phi_{k}(\gamma,h)=\emptyset$. Then the number $\ell_{k_{G}}(\gamma,h)$ is
the $h$-coarse quasihyperbolic length of $\gamma$.

If $X$ is $c$-quasiconvex, then $\ell_{k_{G}}(\gamma,0)=\ell_{k_{G}}(\gamma)$
(see, e.g., [5, Proposition A.7 and Remark A.13] and [23, Lemma 2.5] ).

######  Definition 5.

Let $G$ be a proper domain in a rectifiably connected metric space $X$. An arc
$\gamma\subset G$ is $(\nu,h)$-solid with $\nu\geq 1$ and $h\geq 0$ if

$\ell_{k_{G}}(\gamma[x,y],h)\leq\nu\;k_{G}(x,y)$

for all $x$, $y\in\gamma$.

An arc $\gamma\subset G$ with endpoints $x$ and $y$ is said to be
$\varepsilon$-short ($\varepsilon\geq 0$) if

$\ell_{k_{G}}(\gamma)\leq k_{G}(x,y)+\varepsilon.$

Obviously, by the definition of $k_{G}$, we know that for every
$\varepsilon>0$, there exists an arc $\gamma\subset G$ such that $\gamma$ is
$\varepsilon$-short, and it is easy to see that every subarc of an
$\varepsilon$-short arc is also $\varepsilon$-short.

######  Remark 5.

For any pair of points $x$ and $y$ in a proper domain $G$ of Banach space $E$,
if the dimension of $E$ is finite, then there exists a quasihyperbolic
geodesic in $G$ connecting $x$ and $y$ (see [12, Lemma 1]). But if the
dimension of $E$ is infinite, this property is no longer valid (see, e.g.,
[36, Example 2.9]). In order to overcome this shortcoming in Banach spaces,
Väisälä proved the existence of neargeodesics or quasigeodesics (see [37]),
and every quasihyperbolic geodesic is a quasigeodesic. See also [31]. In
metric spaces, we do not know if this existence property is true or not.
However, this existence property plays a very important role in the related
discussions. In order to overcome this disadvantage, in this paper, we will
exploit the substitution of “quasigeodesics” replaced by “short arcs”. The
class of short arcs has been introduced when Väisälä studied properties of
Gromov hyperbolic spaces [42] (see also [6, 19]), and as we see that the
existence of such class of arcs is obvious in metric spaces. Although, there
is no implication between the class of quasigeodesics and the one of short
arcs, we will prove that, under certain geometric assumptions, every short arc
is a double cone arc.

By a slight modification of the method used in the proof of [37, Lemma 6.21],
we get the following result.

######  Lemma 1.

Suppose that $X$ is a $c$-quasiconvex metric space and that $G\varsubsetneq X$
is a domain, and that $\gamma$ is a $(\nu,h)$-solid arc in $G$ with endpoints
$x$, $y$ such that $\min\\{\delta_{G}(x),\delta_{G}(y)\\}=r\geq 3c|x-y|$. Then
there is a constant $\mu_{1}=\mu_{1}(c,\nu)$ such that

${\operatorname{diam}}(\gamma)\leq\max\\{\mu_{1}|x-y|,2r(e^{h}-1)\\},$

where “${\operatorname{diam}}$” means “diameter”.

Proof. Without loss of generality, we assume that
$\delta_{G}(y)\geq\delta_{G}(x)=r$. Denoting $t=|x-y|$ and applying Lemma , we
get

$k_{G}(x,y)\leq 3ct/r.$

Let $u\in\gamma$. To prove this lemma, it suffices to show that there exists a
constant $\mu_{1}=\mu_{1}(c,\nu)$ such that

(2.2) $|u-x|\leq\max\big{\\{}\frac{\mu_{1}}{2}|x-y|,r(e^{h}-1)\big{\\}}.$

To this end, we consider two cases. The first case is: $k_{G}(u,x)\leq h$.
Under this assumption, it is easy to see that

(2.3) $|u-x|\leq(e^{k_{G}(u,x)}-1)\delta_{G}(x)\leq r(e^{h}-1).$

For the remaining case: $k_{G}(u,x)>h$, we choose a sequence of successive
points of $\gamma$: $x=x_{0}$, $\ldots$, $x_{n}=u$ such that

$k_{G}(x_{j-1},x_{j})=h\;\;\;{\rm for}\;\;\;j\in\\{1,\ldots,n-1\\}$

and

$0<k_{G}(x_{n-1},x_{n})\leq h.$

Then $n\geq 2$ and

$(n-1)h\leq\sum_{i=1}^{n-1}k_{G}(x_{j-1},x_{j})\leq\ell_{k_{G}}(\gamma,h)\leq\nu
k_{G}(x,y)\leq 3c\nu t/r,$

which shows that

$k_{G}(x,u)\leq\sum_{i=1}^{n}k_{G}(x_{j-1},x_{j})\leq nh\leq 6c\nu t/r.$

Let $s=\frac{t}{r}$. Then $s\leq\frac{1}{3c}$ and

$\frac{|u-x|}{t}\leq\frac{e^{6c\nu s}-1}{s}.$

Obviously, the function $g(s)=\frac{1}{s}\big{(}e^{6c\nu s}-1\big{)}$ is
increasing in $(0,\frac{1}{3c}]$ and $\lim_{s\to 0}{\frac{e^{6c\nu
s}-1}{s}}=6c\nu$. Letting

$\mu_{1}=6c(e^{2\nu}-1)$

gives

(2.4) $|u-x|\leq\frac{1}{2}\mu_{1}t.$

It follows from (2.3) and (2.4) that (2.2) holds, and hence the proof of the
lemma is complete. ∎

######  Lemma 2.

Suppose that $X$ is a $c$-quasiconvex metric space and $G\subsetneq X$ is a
domain. Suppose, further, that for $x$, $y\in G$,

  1. (1)

$\gamma$ is an $\varepsilon$-short arc in $G$ connecting $x$ and $y$ with
$0<\varepsilon\leq\frac{1}{2}k_{G}(x,y)$, and

  2. (2)

$|x-y|\leq\frac{1}{3c}\min\\{\delta_{G}(x),\delta_{G}(y)\\}$.

Then

$\ell(\gamma)\leq\frac{9}{2}ce^{\frac{3}{2}}|x-y|.$

Proof. Without loss of generality, we assume that
$\min\\{\delta_{G}(x),\delta_{G}(y)\\}=\delta_{G}(x)$. It follows from Lemma
that

$\log\left(1+\frac{\ell(\gamma)}{\delta_{G}(x)}\right)\leq\ell_{k_{G}}(\gamma)\leq
k_{G}(x,y)+\varepsilon\leq\frac{3}{2}k_{G}(x,y)\leq\frac{9c}{2}\frac{|x-y|}{\delta_{G}(x)}\leq\frac{3}{2}.$

Hence,

$\frac{\ell(\gamma)}{\delta_{G}(x)}\leq e^{\frac{3}{2}}-1,$

which leads to

$\frac{\ell(\gamma)}{\delta_{G}(x)}\leq
e^{\frac{3}{2}}\log\left(1+\frac{\ell(\gamma)}{\delta_{G}(x)}\right)\leq\frac{9}{2}ce^{\frac{3}{2}}\frac{|x-y|}{\delta_{G}(x)}.$

Therefore,

$\ell(\gamma)\leq\frac{9}{2}ce^{\frac{3}{2}}|x-y|,$

as required.∎

###  2.2. Uniform domains, John domains and locally John domains

In 1961, John [26] introduced the twisted interior cone condition with his
work on elasticity, and these domains where first called John domains by
Martio and Sarvas in [30]. In the same paper, Martio and Sarvas also
introduced another class of domains which are the uniform domains. The main
motivation for studying these domains was in showing global injectivity
properties for locally injective mappings. Since then, many other
characterizations of uniform and John domains have been established, see [10,
12, 29, 37, 40, 41], and the importance of these classes of domains in the
function theory is well documented (see e.g. [10, 34]). Moreover, John and
uniform domains in $\mathbb{R}^{n}$ enjoy numerous geometric and function
theoretic properties that are useful in other many fields of modern
mathematical analysis as well (see e.g. [1, 3, 12, 14, 20, 27, 28, 34], and
references therein).

We recall the definitions of uniform domains and John domains following
closely the notation and terminology of [32, 34, 35, 36, 37] and [29].

######  Definition 6.

A domain $G$ in $X$ is called $b$-uniform provided there exists a constant $b$
with the property that each pair of points $x$, $y$ in $G$ can be joined by a
rectifiable arc $\gamma$ in $G$ satisfying

  1. (1)

$\min\\{\ell(\gamma[x,z]),\ell(\gamma[z,y])\\}\leq b\,\delta_{G}(z)$ for all
$z\in\gamma$, and

  2. (2)

$\ell(\gamma)\leq b\,|x-y|$,

where $\ell(\gamma)$ denotes the length of $\gamma$, $\gamma[x,z]$ the part of
$\gamma$ between $x$ and $z$. At this time, $\gamma$ is said to be a double
$b$-cone arc. Condition $(1)$ (resp. $(2)$) is called the cigar condition
(resp. turning condition).

If the condition $(1)$ is satisfied, not necessarily $(2)$, then $G$ is said
to be a $b$-John domain. At this time, the arc $\gamma$ is called a $b$-cone
arc.

######  Definition 7.

A domain $G$ in $X$ is said to be a locally $a$-John domain if there exists a
constant $a$ such that all metric balls $\mathbb{B}(x,r)$ are $a$-John
domains, where $x\in G$ and $0<r\leq\delta_{G}(x)$.

We note that all domains which satisfies the strong geodesic condition (see
[8] for the definition) in the abstract setting of homogeneous spaces are
locally John domains [8, Corollary 3.2]. In particular, all domains in Carnot-
Carathéodory metric spaces [8] and Banach spaces [37] are locally John
domains.

We remark that for $x\in G$, it is possible that the metric ball
$\mathbb{B}(x,r)$ is not contained in $G$ for $0<r\leq\delta_{G}(x)$. But, in
[23], the authors proved the following.

Lemma C. Suppose $X$ is a $c$-quasiconvex metric space and $G\subsetneq X$ is
a domain. Then for any rectifiably connected open set
$D\subset\mathbb{B}(x,r)$ with $x\in D\cap G$, if $r\leq\delta_{G}(x)$, then
$D\subset G.$

It follows from Lemma that the following result is obvious.

######  Lemma 3.

Suppose $X$ is a $c$-quasiconvex metric space and $G\subsetneq X$ is a locally
John domain. For any $x\in G$, if $0<r\leq\delta_{G}(x)$, then
$\mathbb{B}(x,r)\subset G$.

Further, for locally John domains, we have the following estimate for the
quasihyperbolic metric.

######  Lemma 4.

Suppose $G\varsubsetneq X$ is a locally $a$-John domain. Then for $x,$ $y\in
G$ with $|x-y|=t\delta_{G}(x)$, where $0<t<1$, we have

$k_{G}(x,y)\leq 2a\frac{3+t}{1-t}.$

Proof. Let $x,$ $y\in G$ with $|x-y|=t\delta_{G}(x)$. Since $G$ is locally
$a$-John, we know from Definition 7 and Lemma 4 that
$\mathbb{B}(x,\frac{1+t}{2}\delta_{G}(x))\subset G$ is an $a$-John domain.
Then there is a curve $\gamma\in\mathbb{B}(x,\frac{1+t}{2}\delta_{G}(x))$ such
that for any $w\in\gamma$,

$\min\\{\ell(\gamma[y,w]),\ell(\gamma[x,w])\\}\leq a\delta_{G}(w)$

and

$\delta_{G}(w)\geq\delta_{G}(x)-|x-w|\geq\frac{1}{2}(1-t)\delta_{G}(x).$

Let $x_{0}\in\gamma$ be the point bisecting the arc length of $\gamma$. Then

$\ell(\gamma)\leq 2a\delta_{G}(x_{0})\leq 2a(\delta_{G}(x)+|x-x_{0}|)\leq
a(3+t)\delta_{G}(x),$

which shows that

$k_{G}(x,y)\leq\int_{\gamma}\frac{|dw|}{\delta_{G}(w)}\leq\frac{2\ell(\gamma)}{(1-t)\delta_{G}(x)}\leq
2a\frac{3+t}{1-t},$

as required. ∎

Let us recall the following useful property of uniform domains.

Lemma D. $($[5, Lemma 3.12]$)$ Suppose $G\subsetneq X$ is a $b$-uniform domain
in a rectifiable connected metric space $X$. Then for any $x,y\in G$, we have

$k_{G}(x,y)\leq
4b^{2}\log\left(1+\frac{|x-y|}{\min\\{\delta_{G}(x),\delta_{G}(y)\\}}\right).$

The following are the analogues of Lemmas $6.10$ and $6.11$ in [37] in the
setting of metric spaces. The proofs are similar.

Lemma E. Suppose that $G\subsetneq X$ is a $b$-uniform domain in a rectifiable
connected metric space $X$, and that $\gamma$ is an arc in $\\{x\in
G:\delta_{G}(x)\leq r\\}$. If $\gamma$ is $(\nu,h)$-solid, then

${\operatorname{diam}}(\gamma)\leq M_{1}r,$

where $M_{1}=M_{1}(b,\nu,h)$.

Lemma F. For all $b\geq 1$, $\nu\geq 1$ and $h\geq 0$, there are constants
$0<q_{0}=q_{0}(b,\nu,h)<1$ and $M_{2}=M_{2}(b,\nu,h)\geq 1$ with the following
property: Suppose that $G$ is a $b$-uniform domain and $\gamma$ is a
$(\nu,h)$-solid arc starting at $x_{0}\in G$. If $\gamma$ contains a point $u$
with $\delta_{G}(u)\leq q_{0}\delta_{G}(x_{0})$, then

${\operatorname{diam}}(\gamma_{u})\leq M_{2}\delta_{G}(u),$

where $\gamma_{u}=\gamma\setminus\gamma[x_{0},u)$.

Now, we are ready to prove an analogue of Lemma 1 for uniform domains.

######  Lemma 5.

Suppose that $X$ is a $c$-quasiconvex metric space and that $G\varsubsetneq X$
is a $b$-uniform domain, and that $\gamma$ is a $(\nu,h)$-solid arc in $G$
with endpoints $x$, $y$. Let
$\delta_{G}(x_{0})=\max_{p\in\gamma}\delta_{G}(p)$. Then there exist constants
$\mu_{2}=\mu_{2}(b,\nu,h)\geq 1$ and $\mu_{3}=\mu_{3}(b,c,\nu,h)\geq 1$ such
that

  1. (1)

${\operatorname{diam}}(\gamma[x,u])\leq\mu_{2}\delta_{G}(u)$ for
$u\in\gamma[x,x_{0}],$ and
${\operatorname{diam}}(\gamma[y,v])\leq\mu_{2}\delta_{G}(v)$ for
$v\in\gamma[y,x_{0}]$;

  2. (2)

${\operatorname{diam}}(\gamma)\leq\max\big{\\{}\mu_{3}|x-y|,2(e^{h}-1)\min\\{\delta_{G}(x),\delta_{G}(y)\\}\big{\\}}.$

Proof. We first prove (1). Obviously, it suffices to prove the first
inequality in (1) because the proof for the second one is similar. Let

$\mu_{2}=\max\Big{\\{}\frac{M_{1}}{q_{0}},M_{2}\Big{\\}},$

where $M_{1}=M_{1}(b,\nu,h)$ is the constant from Lemma ,
$q_{0}=q_{0}(b,\nu,h)$ and $M_{2}=M_{2}(b,\nu,h)$ are the constants from Lemma
.

For $u\in\gamma[x,x_{0}]$, we divide the proof into two cases. If
$\delta_{G}(u)\leq q_{0}\delta_{G}(x_{0})$, then Lemma leads to

(2.5) ${\operatorname{diam}}(\gamma[x,u])\leq M_{2}\delta_{G}(u).$

If $\delta_{G}(u)>q_{0}\delta_{G}(x_{0})$, then applying Lemma with the
substitution $r$ replaced by $\delta_{G}(x_{0})$ and $\gamma$ replaced by
$\gamma[x,u]$, we easily get

(2.6) ${\operatorname{diam}}(\gamma[x,u])\leq
M_{1}\delta_{G}(x_{0})<\frac{M_{1}}{q_{0}}\delta_{G}(u).$

It follows from (2.5) and (2.6) that the first assertion in (1) holds, and
thus the proof of (1) is complete.

To prove (2), without loss of generality, we assume that

$\min\\{\delta_{G}(x),\;\delta_{G}(y)\\}=\delta_{G}(x)\;\;\;{\rm
and}\;\;\;{\operatorname{diam}}(\gamma)>|x-y|.$

Let

$\mu_{3}=\frac{3}{4}\big{[}1+2(1+6c)(e^{h+4b^{2}\nu\log(1+4\mu_{2})}-1)\big{]}.$

If $\delta_{G}(x)\geq 3c|x-y|$, then $(2)$ follows from Lemma 1 since the
constant $\mu_{1}$ in Lemma 1 satisfies $\mu_{1}<\mu_{3}$. Hence, in the
following, we assume that

$\delta_{G}(x)<3c|x-y|.$

Let $x_{1}\in\gamma$ (resp. $y_{1}\in\gamma$) be the first point in $\gamma$
from $x$ to $y$ (resp. from $y$ to $x$) such that

(2.7)
$\displaystyle{\operatorname{diam}}(\gamma[x,x_{1}])=\frac{1}{2}|x-y|\;\,(\mbox{resp.}\;{\operatorname{diam}}(\gamma[y,y_{1}])=\frac{1}{2}|x-y|).$

Then we have

${\operatorname{diam}}(\gamma[y,x_{1}])\geq|y-x_{1}|\geq|y-x|-|x-x_{1}|\geq\frac{1}{2}|y-x|={\operatorname{diam}}(\gamma[x,x_{1}]),$

and similarly, we get

${\operatorname{diam}}(\gamma[x,y_{1}])>{\operatorname{diam}}(\gamma[y,y_{1}]).$

Thus, it follows from (1) that

$\frac{1}{2}|x-y|={\operatorname{diam}}(\gamma[x,x_{1}])={\operatorname{diam}}(\gamma[y,y_{1}])\leq\mu_{2}\min\\{\delta_{G}(x_{1}),\delta_{G}(y_{1})\\}.$

Also,

$|x_{1}-y_{1}|\leq|x_{1}-x|+|x-y|+|y-y_{1}|\leq 2|x-y|.$

Then Lemma implies

$k_{G}(x_{1},y_{1})\leq
4b^{2}\log\left(1+\frac{|x_{1}-y_{1}|}{\min\\{\delta_{G}(x_{1}),\delta_{G}(y_{1})\\}}\right)\leq
4b^{2}\log(1+4\mu_{2}).$

Since $\gamma$ is a $(\nu,h)$-solid arc, for any $u_{1},$
$u_{2}\in\gamma[x_{1},y_{1}]$, we have

$\displaystyle k_{G}(u_{1},u_{2})$ $\displaystyle\leq$
$\displaystyle\max\\{h,\;\ell_{k_{G}}(\gamma[x_{1},y_{1}],h)\\}\leq h+\nu
k_{G}(x_{1},y_{1})$ $\displaystyle\leq$ $\displaystyle
h+4b^{2}\nu\log(1+4\mu_{2}),$

and so, for all $z\in\gamma[x_{1},y_{1}]$,

$\displaystyle|z-x_{1}|$ $\displaystyle\leq$
$\displaystyle(e^{k_{G}(z,x_{1})}-1)\delta_{G}(x_{1})$ $\displaystyle\leq$
$\displaystyle(e^{h+4b^{2}\nu\log(1+4\mu_{2})}-1)(\delta_{G}(x)+|x-x_{1}|)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}(1+6c)(e^{h+4b^{2}\nu\log(1+4\mu_{2})}-1)|x-y|.$

Let $w_{1},$ $w_{2}\in\gamma$ be points such that

(2.9) $|w_{1}-w_{2}|\geq\frac{2}{3}{\operatorname{diam}}(\gamma).$

Then we get

######  Claim 2.1.

$|w_{1}-w_{2}|\leq\frac{2}{3}\mu_{3}|x-y|$.

Since (2.7) guarantees that neither $\gamma[x,x_{1}]$ nor $\gamma[y,y_{1}]$
contains the set $\\{w_{1},w_{2}\\}$, we see that, to prove this claim,
according to the positions of $w_{1}$ and $w_{2}$ in $\gamma$, we need to
consider the following four possibilities.

  1. (1)

$w_{1}\in\gamma[x,x_{1}]$ and $w_{2}\in\gamma[y,y_{1}]$. Obviously, by (2.7),
we have

$|w_{1}-w_{2}|\leq|w_{1}-x|+|x-y|+|y-w_{2}|\leq 2|x-y|.$

  2. (2)

$w_{1}\in\gamma[x,x_{1}]$ and $w_{2}\in\gamma[x_{1},y_{1}]$. Then (2.7) and
(2.2) show that

$|w_{1}-w_{2}|\leq|w_{1}-x_{1}|+|x_{1}-w_{2}|\leq\frac{1}{2}\big{[}1+(1+6c)(e^{h+4b^{2}\nu\log(1+4\mu_{2})}-1)\big{]}|x-y|.$

  3. (3)

$w_{1},$ $w_{2}\in\gamma[x_{1},y_{1}]$. Then (2.2) implies

$|w_{1}-w_{2}|\leq|w_{1}-x_{1}|+|x_{1}-w_{2}|\leq(1+6c)(e^{h+4b^{2}\nu\log(1+4\mu_{2})}-1)|x-y|.$

  4. (4)

$w_{1}\in\gamma[x_{1},y_{1}]$ and $w_{2}\in\gamma[y_{1},y]$. Again, we infer
from (2.7) and (2.2) that

$|w_{1}-w_{2}|\leq|w_{1}-x_{1}|+|x_{1}-y_{1}|+|y_{1}-w_{2}|\leq\frac{1}{2}\big{[}1+2(1+6c)(e^{h+4b^{2}\nu\log(1+4\mu_{2})}-1)\big{]}|x-y|.$

The claim is proved.

Now, we are ready to finish the proof. It follows from (2.9) and Claim 2.1
that

${\operatorname{diam}}(\gamma)\leq\frac{3}{2}|w_{1}-w_{2}|\leq\mu_{3}|x-y|,$

which implies that (2) also holds in this case. Hence, the proof of the lemma
is complete. ∎

###  2.3. Quasihyperbolic mappings, coarsely quasihyperbolic mappings and
relative homeomorphisms

######  Definition 8.

Let $G\varsubsetneq X$ and $G^{\prime}\varsubsetneq Y$ be two domains. We say
that a homeomorphism $f:G\to G^{\prime}$ is

  1. (1)

$C$-coarsely $M$-quasihyperbolic, or briefly $(M,C)$-CQH, if there are
constants $M\geq 1$ and $C\geq 0$ such that for all $x$, $y\in G$,

$\frac{k_{G}(x,y)-C}{M}\leq k_{G^{\prime}}(f(x),f(y))\leq M\;k_{G}(x,y)+C.$

  2. (2)

$M$-quasihyperbolic, or briefly $M$-QH, if $f$ is $(M,0)$-CQH.

  3. (3)

fully $C$-coarsely $M$-quasihyperbolic if there are constants $M\geq 1$ and
$C\geq 0$ such that $f$ is $C$-coarsely $M$-quasihyperbolic in every subdomain
of $G$.

Under coarsely quasihyperbolic mappings, we have the following useful
relationship between short arcs and solid arcs.

######  Lemma 6.

Suppose that $X$ and $Y$ are rectifiable connected metric spaces, and that
$G\varsubsetneq X$ and $G^{\prime}\varsubsetneq Y$ are domains. If $f:\;G\to
G^{\prime}$ is $(M,C)$-CQH, and $\gamma$ is an $\varepsilon$-short arc in $G$
with $0<\varepsilon\leq 1$, then there are constants $\nu=\nu(C,M)$ and
$h=h(C,M)$ such that the image $\gamma^{\prime}$ of $\gamma$ under $f$ is
$(\nu,h)$-solid in $G^{\prime}$.

Proof. Let

$h=(2M+1)C+2M\;\;\mbox{and}\;\;\nu=\frac{4(C+1)M(M+1)}{2C+1}.$

Obviously, we only need to verify that for $x$ and $y\in\gamma$,

(2.10) $\ell_{k_{G^{\prime}}}(\gamma^{\prime}[x^{\prime},y^{\prime}],h)\leq\nu
k_{G^{\prime}}(x^{\prime},y^{\prime}).$

We prove this by considering two cases. The first case is: $k_{G}(x,y)<2C+1$.
Then for $z_{1}$, $z_{2}\in\gamma[x,y]$, we have

$k_{G^{\prime}}(z^{\prime}_{1},z^{\prime}_{2})\leq Mk_{G}(z_{1},z_{2})+C\leq
M(k_{G}(x,y)+\varepsilon)+C<(2M+1)C+2M=h,$

and so

(2.11) $\ell_{k_{G^{\prime}}}(\gamma^{\prime}[x^{\prime},y^{\prime}],h)=0.$

Now, we consider the other case: $k_{G}(x,y)\geq 2C+1$. Then

$k_{G^{\prime}}(x^{\prime},y^{\prime})\geq\frac{1}{M}(k_{G}(x,y)-C)>\frac{1}{2M}k_{G}(x,y).$

With the aid of [37, Theorems 4.3 and 4.9], we have

$\displaystyle\ell_{k_{G^{\prime}}}(\gamma^{\prime}[x^{\prime},y^{\prime}],h)$
$\displaystyle\leq$
$\displaystyle\ell_{k_{G^{\prime}}}(\gamma^{\prime}[x^{\prime},y^{\prime}],(M+1)C)\leq(M+1)\ell_{k_{G}}(\gamma[x,y])$
$\displaystyle\leq$ $\displaystyle(M+1)(k_{G}(x,y)+\varepsilon)$
$\displaystyle\leq$ $\displaystyle\frac{2(C+1)(M+1)}{2C+1}k_{G}(x,y)$
$\displaystyle\leq$
$\displaystyle\frac{4(C+1)M(M+1)}{2C+1}k_{G^{\prime}}(x^{\prime},y^{\prime}).$

It follows from (2.11) and (2.3) that (2.10) holds.∎

The following two lemmas are useful in the proof of Theorem 1.

######  Lemma 7.

Suppose that $X$ and $Y$ are both $c$-quasiconvex and complete metric spaces,
and that $G\varsubsetneq X$ and $G^{\prime}\varsubsetneq Y$ are domains. If
both $f:$ $G\to G^{\prime}$ and $f^{-1}:$ $G^{\prime}\to G$ are weakly
$H$-quasisymmetric, then

  1. (1)

$f$ is $\varphi$-FQC, where $\varphi=\varphi_{c,H}$ which means that the
function $\varphi$ depends only on $c$ and $H$;

  2. (2)

$f$ is fully $(M,C)$-CQH, where $M=M(c,H)\geq 1$ and $C=C(c,H)\geq 0$ are
constants.

Proof. By [25, Theorem 1.6], we know that for every subdomain $D\subset G$,
both $f:$ $D\to D^{\prime}$ and $f^{-1}:$ $D^{\prime}\to D$ are
$\varphi$-semisolid with $\varphi=\varphi_{c,H}$, and so, $f$ is
$\varphi$-FQC. Hence (1) holds. Meanwhile, [23, Theorem 1] implies that (1)
and (2) are equivalent, and thus, (2) also holds. ∎

Lemma G. $($[41, Lemma 6.5]$)$ Suppose that $X$ is $c$-quasiconvex, and that
$f:$ $X\to Y$ is weakly $H$-qasisymmetric. If $x,$ $y,$ $z$ are distinct
points in $X$ with $|y-x|\leq t|z-x|$, then

$|y^{\prime}-x^{\prime}|\leq\theta(t)|z^{\prime}-x^{\prime}|,$

where the function $\theta(t)=\theta_{c,H}(t)$ is increasing in $t$.

######  Definition 9.

Let $G\varsubsetneq X$ and $G^{\prime}\varsubsetneq Y$ be two domains. We say
that a homeomorphism $f:G\to G^{\prime}$ is

  1. (1)

$(\theta,t_{0})$-relative if there is a constant $t_{0}\in(0,1]$ and a
homeomorphism $\theta:[0,t_{0})\to[0,\infty)$ such that

$\frac{|x^{\prime}-y^{\prime}|}{\delta_{G^{\prime}}(x^{\prime})}\leq\theta\Big{(}\frac{|x-y|}{\delta_{G}(x)}\Big{)}$

whenever $x$, $y\in G$ and $|x-y|<t_{0}\delta_{G}(x)$; In particular, if
$t_{0}=1$, then $f$ is called to be $\theta$-relative;

  2. (2)

fully $(\theta,t_{0})$-relative (resp. fully $\theta$-relative) if $f$ is
$(\theta,t_{0})$-relative (resp. $\theta$-relative) in every subdomain of $G$.

##  3\. The proof of Theorem 1

In this section, we always assume that $X$ and $Y$ are $c$-quasiconvex and
complete metric spaces, and that $G\varsubsetneq X$ and
$G^{\prime}\varsubsetneq Y$ are domains. Furthermore, we suppose that both
$f:$ $G\to G^{\prime}$ and $f^{-1}:$ $G^{\prime}\to G$ are weakly
$H$-quasisymmetric, $G^{\prime}$ is $c_{1}$-quasiconvex and $D\subset G$ is
$b$-uniform.

Under these assumptions, it follows from Lemma 7 that $f$ is $(M,C)$-$CQH$
with $M=M(c,H)\geq 1$ and $C=C(c,H)\geq 0$.

We are going to show the uniformity of $D^{\prime}=f(D)$. For this, we let
$x^{\prime}$, $y^{\prime}\in D^{\prime}=f(D)\subset G^{\prime}$, and
$\gamma^{\prime}$ be an $\varepsilon$-short arc in $D^{\prime}$ joining
$x^{\prime}$ and $y^{\prime}$ with

$0<\varepsilon<\min\big{\\{}1,\frac{1}{2}k_{D^{\prime}}(x^{\prime},y^{\prime})\big{\\}}.$

Then by Lemmas 6 and 7(2), the preimage $\gamma$ of $\gamma^{\prime}$ is a
$(\nu,h)$-solid arc in $D$ with $\nu=\nu(c,H)$ and $h=h(c,H)$. Let
$w_{0}\in\gamma$ be such that

(3.1) $\delta_{D}(w_{0})=\max_{p\in\gamma}\delta_{D}(p).$

Then by Lemma 5, there is a constant $\mu=\mu(b,\nu,h)$ such that for each
$u\in\gamma[x,w_{0}]$ and for all $z\in\gamma[u,w_{0}]$,

(3.2) $|u-z|\leq{\operatorname{diam}}(\gamma[u,z])\leq\mu\delta_{D}(z),$

and for each $v\in\gamma[y,w_{0}]$ and for all $z\in\gamma[v,w_{0}]$,

$|v-z|\leq{\operatorname{diam}}(\gamma[v,z])\leq\mu\delta_{D}(z).$

In the following, we show that $\gamma^{\prime}$ is a double cone arc in
$D^{\prime}$. Precisely, we shall prove that there exist constants $A\geq 1$
and $B\geq 1$ such that for every
$z^{\prime}\in\gamma^{\prime}[x^{\prime},y^{\prime}]$,

(3.3)
$\min\\{\ell(\gamma^{\prime}[x^{\prime},z^{\prime}]),\ell(\gamma^{\prime}[z^{\prime},y^{\prime}])\\}\leq
A\delta_{D^{\prime}}(z^{\prime})$

and

(3.4) $\ell(\gamma^{\prime})\leq B|x^{\prime}-y^{\prime}|.$

The verification of (3.3) and (3.4) is given in the following two subsections.

###  3.1. The proof of (3.3)

Let

$A=2e^{8b^{2}A_{1}(C+1)M}\;\;{\rm
and}\;\;A_{1}=2e^{M+C}(1+\mu)\theta^{\prime\prime}\Big{(}6c\theta^{\prime}(\mu)e^{4b^{2}M+C}\Big{)},$

where the functions $\theta^{\prime}=\theta^{\prime}_{b,H}$ and
$\theta^{\prime\prime}=\theta^{\prime\prime}_{c_{1},H}$ are from Lemma .
Obviously, we only need to get the following estimate: For all
$z^{\prime}\in\gamma^{\prime}[x^{\prime},w^{\prime}_{0}]$ (resp.
$z^{\prime}\in\gamma^{\prime}[y^{\prime},w^{\prime}_{0}]$),

(3.5) $\ell(\gamma^{\prime}[x^{\prime},z^{\prime}])\leq
A\delta_{D^{\prime}}(z^{\prime})\;\;({\rm
resp.}\;\ell(\gamma^{\prime}[y^{\prime},z^{\prime}])\leq
A\delta_{D^{\prime}}(z^{\prime})).$

It suffices to prove the case
$z^{\prime}\in\gamma^{\prime}[x^{\prime},w^{\prime}_{0}]$ since the proof of
the case $z^{\prime}\in\gamma^{\prime}[y^{\prime},w^{\prime}_{0}]$ is similar.
Suppose on the contrary that there exists some point
$x^{\prime}_{0}\in\gamma^{\prime}[x^{\prime},w^{\prime}_{0}]$ such that

(3.6)
$\ell(\gamma^{\prime}[x^{\prime},x_{0}^{\prime}])>A\delta_{D^{\prime}}(x_{0}^{\prime}).$

Then we choose $x^{\prime}_{1}\in\gamma^{\prime}[x^{\prime},w^{\prime}_{0}]$
be the first point from $x^{\prime}$ to $w_{0}^{\prime}$ such that

(3.7)
$\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{1}])=A\delta_{D^{\prime}}(x^{\prime}_{1}).$

Let $x_{2}\in D$ be such that

$|x_{1}-x_{2}|=\frac{1}{2}\delta_{D}(x_{1}).$

Then we have

######  Claim 3.1.

$|x_{1}^{\prime}-x_{2}^{\prime}|<e^{4b^{2}M+C}\delta_{D^{\prime}}(x_{1}^{\prime}).$

Obviously,

$\delta_{D}(x_{2})\geq\delta_{D}(x_{1})-|x_{1}-x_{2}|=|x_{1}-x_{2}|,$

and so, Lemma implies

$\displaystyle\log\left(1+\frac{|x_{1}^{\prime}-x_{2}^{\prime}|}{\delta_{D^{\prime}}(x_{1}^{\prime})}\right)$
$\displaystyle\leq$ $\displaystyle
k_{D^{\prime}}(x_{1}^{\prime},x_{2}^{\prime})\leq Mk_{D}(x_{1},x_{2})+C$
$\displaystyle\leq$ $\displaystyle
4b^{2}M\log\left(1+\frac{|x_{1}-x_{2}|}{\min\\{\delta_{D}(x_{1}),\delta_{D}(x_{2})\\}}\right)+C$
$\displaystyle<$ $\displaystyle 4b^{2}M+C,$

whence

$|x_{1}^{\prime}-x_{2}^{\prime}|<e^{4b^{2}M+C}\delta_{D^{\prime}}(x_{1}^{\prime}),$

which shows that the claim holds.

Let $x^{\prime}_{3}\in\gamma^{\prime}[x^{\prime},x^{\prime}_{1}]$ be such that

(3.8)
$\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{3}])=\frac{1}{2}\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{1}]),$

and then, we get an estimate on $|x_{1}^{\prime}-x_{2}^{\prime}|$ in terms of
$d_{D^{\prime}}(x_{3}^{\prime})$ as stated in the following claim.

######  Claim 3.2.

$|x_{1}^{\prime}-x_{2}^{\prime}|<2e^{4b^{2}M+C}\delta_{D^{\prime}}(x_{3}^{\prime}).$

It follows from (3.7) and (3.8) that

$\delta_{D^{\prime}}(x_{1}^{\prime})<2\delta_{D^{\prime}}(x_{3}^{\prime}),$

since the choice of $x_{1}^{\prime}$ implies
$\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{3}])<Ad_{D^{\prime}}(x_{3}^{\prime})$.
And so, Claim 3.1 leads to

$|x_{1}^{\prime}-x_{2}^{\prime}|<2e^{4b^{2}M+C}\delta_{D^{\prime}}(x_{3}^{\prime}),$

as required.

Based on Claim 3.2, we have

######  Claim 3.3.

$|x^{\prime}_{1}-x^{\prime}_{3}|\leq
2\theta^{\prime}(\mu)e^{4b^{2}M+C}\delta_{D^{\prime}}(x^{\prime}_{3}).$

In order to exploit Lemma to show this claim, we need some preparation. It
follows from (3.7) and (3.8) that

$\displaystyle k_{D^{\prime}}(x^{\prime}_{1},x^{\prime}_{3})$
$\displaystyle\geq$
$\displaystyle\ell_{k_{D^{\prime}}}(\gamma^{\prime}[x^{\prime}_{1},x^{\prime}_{3}])-\varepsilon\geq\log\Big{(}1+\frac{\ell(\gamma^{\prime}[x^{\prime}_{1},x^{\prime}_{3}])}{\delta_{D^{\prime}}(x^{\prime}_{1})}\Big{)}-1$
$\displaystyle=$ $\displaystyle\log\big{(}1+\frac{A}{2}\big{)}-1.$

Hence, by Lemma , we have

$\displaystyle\log\left(1+\frac{|x_{1}-x_{3}|}{\min\\{\delta_{D}(x_{1}),\delta_{D}(x_{3})\\}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4b^{2}}k_{D}(x_{1},x_{3})\geq\frac{1}{4b^{2}M}(k_{D^{\prime}}(x^{\prime}_{1},x^{\prime}_{3})-C)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4b^{2}M}\Big{(}\log\big{(}1+\frac{A}{2}\big{)}-1-C\Big{)}$
$\displaystyle>$ $\displaystyle\log(1+A_{1}),$

and so

(3.9)
$|x_{1}-x_{3}|>A_{1}\min\\{\delta_{D}(x_{1}),\delta_{D}(x_{3})\\}>\frac{A_{1}}{1+\mu}\delta_{D}(x_{3}),$

since (3.2) implies

$\delta_{D}(x_{3})\leq\delta_{D}(x_{1})+|x_{1}-x_{3}|\leq(1+\mu)\delta_{D}(x_{1}).$

Again, by (3.2), we know

$|x_{1}-x_{3}|\leq\mu\delta_{D}(x_{1})=2\mu|x_{1}-x_{2}|.$

Now, we are ready to apply Lemma to the points $x_{1}$, $x_{2}$ and $x_{3}$ in
$D$. Since $f$ is weakly $H$-quasisymmetric and $D$ is $b$-uniform, by
considering the restriction $f|_{D}$ of $f$ onto $D^{\prime}$, we know from
Lemma that there is an increasing function
$\theta^{\prime}=\theta^{\prime}_{b,H}$ such that

$|x^{\prime}_{1}-x^{\prime}_{3}|\leq\theta^{\prime}(2\mu)|x^{\prime}_{1}-x^{\prime}_{2}|,$

and thus, Claim 3.2 assures that

$|x^{\prime}_{1}-x^{\prime}_{3}|\leq
2\theta^{\prime}(2\mu)e^{4b^{2}M+C}\delta_{D^{\prime}}(x^{\prime}_{3}),$

which completes the proof of Claim 3.3.

Let us proceed the proof. To get a contradiction to the contrary assumption
(3.6), we choose $x^{\prime}_{4}\in D^{\prime}$ such that

(3.10)
$|x^{\prime}_{3}-x^{\prime}_{4}|=\frac{1}{3c}\delta_{D^{\prime}}(x^{\prime}_{3}).$

Then Lemma implies that

$\displaystyle\log\left(1+\frac{|x_{3}-x_{4}|}{\delta_{D}(x_{3})}\right)$
$\displaystyle\leq$ $\displaystyle k_{D}(x_{3},x_{4})\leq
Mk_{D^{\prime}}(x_{3}^{\prime},x_{4}^{\prime})+C$ $\displaystyle\leq$
$\displaystyle
3cM\frac{|x_{3}^{\prime}-x_{4}^{\prime}|}{\delta_{D^{\prime}}(x_{3}^{\prime})}+C\leq
M+C,$

which yields that

(3.11) $|x_{3}-x_{4}|<e^{M+C}\delta_{D}(x_{3}).$

Meanwhile, Claim 3.3 and (3.10) imply that

$|x_{1}^{\prime}-x_{3}^{\prime}|\leq
2\theta^{\prime}(2\mu)e^{4b^{2}M+C}\delta_{D^{\prime}}(x_{3}^{\prime})=6c\theta^{\prime}(2\mu)e^{4b^{2}M+C}|x_{3}^{\prime}-x_{4}^{\prime}|.$

Now, we apply Lemma to the points $x_{1}^{\prime}$, $x_{3}^{\prime}$ and
$x_{4}^{\prime}$ in $G^{\prime}$. Since $f^{-1}:$ $G^{\prime}\to G$ is weakly
$H$-quasisymmetric and $G^{\prime}$ is $c_{1}$-quasiconvex, we know from Lemma
that there is an increasing function
$\theta^{\prime\prime}=\theta^{\prime\prime}_{c_{1},H}$ such that

$|x_{1}-x_{3}|\leq\theta^{\prime\prime}\Big{(}6c\theta^{\prime}(2\mu)e^{4b^{2}M+C}\Big{)}|x_{3}-x_{4}|,$

which, together with (3.9) and (3.11), shows that

$\displaystyle|x_{1}-x_{3}|$ $\displaystyle\leq$ $\displaystyle
e^{M+C}\theta^{\prime\prime}\Big{(}6c\theta^{\prime}(2\mu)e^{4b^{2}M+C}\Big{)}\delta_{D}(x_{3})$
$\displaystyle\leq$
$\displaystyle\frac{1+\mu}{A_{1}}e^{M+C}\theta^{\prime\prime}\Big{(}6c\theta^{\prime}(2\mu)e^{4b^{2}M+C}\Big{)}|x_{1}-x_{3}|$
$\displaystyle=$ $\displaystyle\frac{1}{2}|x_{1}-x_{3}|.$

This obvious contradiction shows that (3.3) is true. ∎

###  3.2. The proof of (3.4)

Let

$B=12cA^{2}e^{6b^{2}M\mu\big{(}1+\theta^{\prime\prime}\big{(}\frac{1+12cA}{3c}\big{)}\big{)}},$

and suppose on the contrary that

(3.12) $\ell(\gamma^{\prime})>B|x^{\prime}-y^{\prime}|.$

Since $\frac{9}{2}ce^{\frac{3}{2}}<B$, we see from Lemma 2 that

(3.13)
$|x^{\prime}-y^{\prime}|>\frac{1}{3c}\max\\{\delta_{D^{\prime}}(x^{\prime}),\delta_{D^{\prime}}(y^{\prime})\\}.$

For convenience, in the following, we assume that

$\max\\{\delta_{D^{\prime}}(x^{\prime}),\delta_{D^{\prime}}(y^{\prime})\\}=\delta_{D^{\prime}}(x^{\prime}).$

First, we choose some special points from $\gamma^{\prime}$. By (3.12), we
know that there exist $w^{\prime}_{1}$ and $w^{\prime}_{2}\in\gamma^{\prime}$
such that $x^{\prime}$, $w^{\prime}_{1}$, $w^{\prime}_{2}$ and $y^{\prime}$
are successive points in $\gamma^{\prime}$ and

(3.14)
$\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{1}])=\ell(\gamma^{\prime}[w^{\prime}_{2},y^{\prime}])=6cA|x^{\prime}-y^{\prime}|.$

Then we have

######  Claim 3.4.

$|x^{\prime}-w^{\prime}_{1}|\geq\frac{1}{2}\delta_{D^{\prime}}(w^{\prime}_{1})$
and
$|y^{\prime}-w^{\prime}_{2}|\geq\frac{1}{2}\delta_{D^{\prime}}(w^{\prime}_{2})$.

Obviously, it suffices to show the first inequality in the claim. Suppose

$|x^{\prime}-w^{\prime}_{1}|<\frac{1}{2}\delta_{D^{\prime}}(w^{\prime}_{1}).$

Then (3.3) and (3.13) lead to

$\delta_{D^{\prime}}(x^{\prime})\geq\delta_{D^{\prime}}(w^{\prime}_{1})-|x^{\prime}-w^{\prime}_{1}|>\frac{1}{2}\delta_{D^{\prime}}(w^{\prime}_{1})\geq\frac{1}{2A}\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{1}])=3c|x^{\prime}-y^{\prime}|>\delta_{D^{\prime}}(x^{\prime}).$

This obvious contradiction completes the proof of Claim 3.4.

By using Claim 3.4, we get a lower bound for $|w_{1}-w_{2}|$ in terms of
$\min\\{\delta_{D}(w_{1}),\;\delta_{D}(w_{2})\\}$, which is as follows.

######  Claim 3.5.

$|w_{1}-w_{2}|>\Big{(}1+\theta^{\prime\prime}\Big{(}\frac{1+12cA}{3c}\Big{)}\Big{)}\mu\min\\{\delta_{D}(w_{1}),\;\delta_{D}(w_{2})\\}.$

Without loss of generality, we assume that
$\min\\{\delta_{D}(w_{1}),\;\delta_{D}(w_{2})\\}=\delta_{D}(w_{1})$. Then by
(3.14) and Claim 3.4, we have

(3.15) $\delta_{D^{\prime}}(w^{\prime}_{1})\leq
2|x^{\prime}-w^{\prime}_{1}|\leq
2\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{1}])=12cA|x^{\prime}-y^{\prime}|.$

Since $\gamma^{\prime}$ is an $\varepsilon$-short arc and $D$ is $b$-uniform,
by Lemma , we have

$\displaystyle\log\left(1+\frac{|w_{1}-w_{2}|}{\delta_{D}(w_{1})}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4b^{2}}k_{D}(w_{1},w_{2})\geq\frac{1}{4Mb^{2}}k_{D^{\prime}}(w^{\prime}_{1},w^{\prime}_{2})-\frac{C}{4Mb^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4Mb^{2}}\ell_{k_{D^{\prime}}}(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])-\frac{\varepsilon+C}{4Mb^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4Mb^{2}}\log\left(1+\frac{\ell(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])}{\delta_{D^{\prime}}(w^{\prime}_{1})}\right)-\frac{1+C}{4Mb^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4Mb^{2}}\log\Big{(}1+\frac{B-12cA}{12cA}\Big{)}-\frac{1+C}{4Mb^{2}}$
$\displaystyle=$ $\displaystyle\lambda,$

where the last inequality follows from (3.15) and the following inequalities:

$\ell(\gamma^{\prime}[w_{1}^{\prime},w_{2}^{\prime}])=\ell(\gamma^{\prime})-\ell(\gamma^{\prime}[x^{\prime},w_{1}^{\prime}])-\ell(\gamma^{\prime}[y^{\prime},w_{2}^{\prime}])>(B-12cA)|x^{\prime}-y^{\prime}|.$

Hence

$|w_{1}-w_{2}|\geq(e^{\lambda}-1)\delta_{D}(w_{1})>\Big{(}1+\theta^{\prime\prime}\Big{(}\frac{1+12cA}{3c}\Big{)}\Big{)}\mu\delta_{D}(w_{1}),$

as required.

Next, we get the following upper bound for $|w_{1}-w_{2}|$ in terms of
$\min\\{\delta_{D}(w_{1}),\;\delta_{D}(w_{2})\\}$.

######  Claim 3.6.

$|w_{1}-w_{2}|\leq\theta^{\prime\prime}\left(\frac{1+12cA}{3c}\right)\mu\min\\{\delta_{D}(w_{1}),\;\delta_{D}(w_{2})\\}.$

First, we see that $w_{0}\in\gamma[w_{1},y]$, where $w_{0}$ is the point in
$\gamma$ which satisfies (3.1), because otherwise (3.2) gives that

$|w_{1}-w_{2}|\leq\mu\delta_{D}(w_{1}),$

which contradicts with Claim 3.5.

We are going to apply Lemma to the points $x^{\prime}$, $w_{1}^{\prime}$ and
$w_{2}^{\prime}$ in $G^{\prime}$. We need a relationship between
$|w^{\prime}_{1}-w^{\prime}_{2}|$ and $|x^{\prime}-w_{1}^{\prime}|$. To this
end, it follows from (3.14) that

$|w^{\prime}_{1}-w^{\prime}_{2}|\leq|w^{\prime}_{1}-x^{\prime}|+|x^{\prime}-y^{\prime}|+|y^{\prime}-w^{\prime}_{2}|\leq(1+12cA)|x^{\prime}-y^{\prime}|\leq\frac{1+12cA}{3c}|x^{\prime}-w_{1}^{\prime}|,$

since we infer from the choice of $w^{\prime}_{1}$, (3.3) and Claim 3.4 that

$|x^{\prime}-w_{1}^{\prime}|\geq\frac{1}{2}\delta_{D^{\prime}}(w_{1}^{\prime})\geq\frac{1}{2A}\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{1}])=3c|x^{\prime}-y^{\prime}|.$

Then by Lemma , we have known that there is an increasing function
$\theta^{\prime\prime}=\theta^{\prime\prime}_{c_{1},H}$ such that

$|w_{1}-w_{2}|\leq\theta^{\prime\prime}\left(\frac{1+12cA}{3c}\right)|x-w_{1}|,$

and thus, (3.2) leads to

$|w_{1}-w_{2}|\leq\theta^{\prime\prime}\left(\frac{1+12cA}{3c}\right)\mu\min\\{\delta_{D}(w_{1}),\;\delta_{D}(w_{2})\\},$

which shows that Claim 3.6 holds.

It follows from Claims 3.5 and 3.6 that it is impossible, and so this obvious
contradiction completes the proof of (3.4).∎

Inequalities (3.3) and (3.4), together with the arbitrariness of the choice of
$x^{\prime}$ and $y^{\prime}$ in $D^{\prime}$, show that $D^{\prime}$ is
$B$-uniform, which implies that Theorem 1 holds.∎

##  4\. The proof of Theorem 2

In this section, we always assume that $X$ and $Y$ are both $c$-quasiconvex
and complete metric spaces, that $G\varsubsetneq X$ is a non-point-cut and
locally $a$-John domain, and that $G^{\prime}\varsubsetneq Y$ is a
$b_{1}$-uniform and locally $a$-John domain. Further, we assume that $f:$
$G\to G^{\prime}$ is a $\varphi$-FQC mapping and $D$ is a $b_{2}$-uniform
subdomain of $G$.

We divide this section into two subsections. In the first subsection, a useful
lemma will be proved, and the proof of Theorem 2 will be presented in the
second subsection.

###  4.1. An auxiliary result

First, based on [23, Theorems 1 and 2], we prove the following result which
plays a key role in the proof of Theorem 2.

######  Lemma 8.

Under the given assumptions in the first paragraph of this section, we have
the following assertions.

  1. (1)

There exist constants $M=M(c,\varphi)\geq 1$ and $C=C(c,\varphi)\geq 0$ such
that both $f:$ $G\to G^{\prime}$ and $f^{-1}:$ $G^{\prime}\to G$ are fully
$(M,C)$-CQH.

  2. (2)

There exists a constant $q=q(c)\in(0,1)$ such that for any $x\in G$, $f$ is
$\eta$-quasisymmetric in $\mathbb{B}(x,q\delta_{G}(x))$ and $f^{-1}$ is
$\eta$-quasisymmetric in
$\mathbb{B}(x^{\prime},q\delta_{G^{\prime}}(x^{\prime}))$, where
$\eta=\eta_{a,c,\varphi}$.

Proof. Since $f$ is a $\varphi$-FQC mapping, we see that Lemma 8(1) easily
follows from [23, Theorem 1]. For the proof of the second assertion, we infer
from [23, Theorem 2] that we only need to prove that there is a homeomorphism
$\theta:[0,1)\to[0,\infty)$ such that both $f$ and $f^{-1}$ are
$\theta$-relative. By symmetry, we know that we only need to show that $f$ is
$\theta$-relative. To this end, we let $0<t<1$ and $x,y\in G$ with
$|x-y|=t\delta_{G}(x)$. Then we separate the proof into two cases. For the
first case, that is, $0<t\leq\frac{1}{3c}$, it follows from Lemma that

$k_{G}(x,y)\leq 3c\frac{|x-y|}{\delta_{G}(x)}=3ct.$

Hence,

(4.1)
$\displaystyle\frac{|x^{\prime}-y^{\prime}|}{\delta_{G^{\prime}}(x^{\prime})}\leq
e^{k_{G^{\prime}}(x^{\prime},y^{\prime})}-1\leq e^{\varphi(k_{G}(x,y))}-1\leq
e^{\varphi(3ct)}-1.$

For the other case, that is, $\frac{1}{3c}<t<1$, by Lemma 4, we know that

$k_{G}(x,y)\leq 2a\frac{3+t}{1-t},$

which implies

(4.2)
$\displaystyle\frac{|x^{\prime}-y^{\prime}|}{\delta_{G^{\prime}}(x^{\prime})}\leq
e^{k_{G^{\prime}}(x^{\prime},y^{\prime})}-1\leq e^{\varphi(k_{G}(x,y))}-1\leq
e^{\varphi\big{(}2a\frac{3+t}{1-t}\big{)}}-1.$

Therefore, (4.1) and (4.2) show that $f$ is $\theta$-relative, where

$\theta(t)=\begin{cases}\displaystyle
e^{\varphi\big{(}\frac{6ac(9c+1)}{3c-1}t\big{)}}-1,\;\mbox{if}\;\;t\in[0,\frac{1}{3c}],\\\
\displaystyle\;\;\;e^{\varphi\big{(}2a\frac{3+t}{1-t})}-1,\;\;\;\,\mbox{if}\;\;t\in(\frac{1}{3c},1).\end{cases}$

Hence the proof of Lemma 8 is complete.∎

###  4.2. The proof of Theorem 2

For $x^{\prime},y^{\prime}\in D^{\prime}$, there is an $\varepsilon$-short arc
$\gamma^{\prime}$ joining $x^{\prime}$ and $y^{\prime}$ in $D^{\prime}$ with

$0<\varepsilon<\frac{1}{2}\min\\{2,k_{D^{\prime}}(x^{\prime},y^{\prime})\\}.$

Lemmas 6 and 8(1) show that the preimage $\gamma$ of $\gamma^{\prime}$ is a
$(\nu,h)$-solid arc in $D$, where $\nu=\nu(c,\varphi)$, $h=h(c,\varphi)$. Let
$z_{0}\in\gamma$ be such that

$\delta_{D}(z_{0})=\max_{p\in\gamma}\delta_{D}(p).$

Then by Lemma 5, there is a constant
$\mu=\mu(b_{2},\nu,h)=\mu(b_{2},c,\varphi)>1$ such that for each
$u\in\gamma[x,z_{0}]$ and for all $z\in\gamma[u,z_{0}]$,

(4.3) $|u-z|\leq\mu\delta_{D}(z),$

and for each $v\in\gamma[y,z_{0}]$ and for all $z\in\gamma[v,z_{0}]$,

(4.4) $|v-z|\leq\mu\delta_{D}(z).$

In the following, we show that $\gamma^{\prime}$ is a double cone arc in
$D^{\prime}$. The proof is divided into two steps which are given in the
following two subsubsections.

####  4.2.1. The verification of the cigar condition

To this end, we let

$\lambda_{1}=\lambda_{2}^{2}(1+\lambda^{4}_{2})^{9(C+1)M^{2}b_{2}^{2}\lambda_{2}}$

and

$\lambda_{2}=\max\Big{\\{}\frac{3c\mu}{\varphi^{-1}(\frac{q}{21ce})},\frac{12c(C+1)\mu}{q},16b_{1}^{2}b_{2}^{2}(C+1)M^{2},\frac{2(1+\mu)}{\eta^{-1}(\frac{1}{3}\eta^{-1}(\frac{1}{\mu}))}\Big{\\}},$

where the constants $C$, $M$, $q$ and the function $\eta$ are from Lemma 8.

In this subsubsection, we show that $\gamma^{\prime}$ satisfies the cigar
condition with constant $2\lambda_{1}^{2}$. Obviously, we may assume that

  1. (1)

either there exists a point
$z_{1}^{\prime}\in\gamma^{\prime}[x^{\prime},z^{\prime}_{0}]$ such that

$\ell(\gamma^{\prime}[x^{\prime},z_{1}^{\prime}])>\lambda_{1}\delta_{D^{\prime}}(z_{1}^{\prime}),$

  2. (2)

or there exists a point
$z_{2}^{\prime}\in\gamma^{\prime}[y^{\prime},z^{\prime}_{0}]$ such that

$\ell(\gamma^{\prime}[y^{\prime},z_{2}^{\prime}])>\lambda_{1}\delta_{D^{\prime}}(z_{2}^{\prime}).$

If (1) happens, then we let
$x^{\prime}_{0}\in\gamma^{\prime}[x^{\prime},z^{\prime}_{0}]$ be the first
point from $x^{\prime}$ to $z^{\prime}_{0}$ such that

(4.5)
$\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{0}])=\lambda_{1}\delta_{D^{\prime}}(x^{\prime}_{0}).$

The following comparison result is useful.

######  Claim 4.1.

For every $z\in\gamma[x_{0},z_{0}]$,
$\delta_{G}(z)\leq\lambda_{2}\delta_{D}(z)$.

Suppose on the contrary that there exists some point
$x_{1}\in\gamma[x_{0},z_{0}]$ with

(4.6) $\delta_{G}(x_{1})>\lambda_{2}\delta_{D}(x_{1}).$

To get a contradiction, we need some preparation. First, it follows from (4.3)
that for all $z\in\gamma[x,x_{1}]$,

$|z-x_{1}|\leq\mu\delta_{D}(x_{1})<\frac{\mu}{\lambda_{2}}\delta_{G}(x_{1})\leq\frac{q}{12c(C+1)}\delta_{G}(x_{1}),$

whence

(4.7)
$\gamma[x,x_{1}]\subset\mathbb{B}\big{(}x_{1},\frac{\mu}{\lambda_{2}}\delta_{G}(x_{1})\big{)}\subset\mathbb{B}\big{(}x_{1},\frac{q}{12c(C+1)}\delta_{G}(x_{1})\big{)}.$

Further, we show that

(4.8)
$\gamma^{\prime}[x^{\prime},x^{\prime}_{1}]\subset\mathbb{B}\big{(}x^{\prime}_{0},\frac{q}{10c}\delta_{G^{\prime}}(x^{\prime}_{0})\big{)}.$

For each $z^{\prime}\in\gamma^{\prime}[x^{\prime},x^{\prime}_{1}]$, by Lemma ,
we have

$k_{G^{\prime}}(z^{\prime},x^{\prime}_{1})\leq\varphi(k_{G}(z,x_{1}))\leq\varphi\big{(}3c\frac{|x_{1}-z|}{\delta_{G}(x_{1})}\big{)}\leq\varphi\big{(}\frac{3c\mu}{\lambda_{2}}\big{)}\leq\frac{q}{21ce},$

and so by the elementary inequality “$e^{x}-1\leq ex$” in $(0,1)$, we get that

$|z^{\prime}-x^{\prime}_{1}|\leq(e^{k_{G^{\prime}}(z^{\prime},x^{\prime}_{1})}-1)\delta_{G^{\prime}}(x^{\prime}_{1})\leq\frac{q}{21c}\delta_{G^{\prime}}(x^{\prime}_{1}),$

which implies

$\delta_{G^{\prime}}(x^{\prime}_{0})\geq\delta_{G^{\prime}}(x^{\prime}_{1})-|x^{\prime}_{0}-x^{\prime}_{1}|\geq\frac{19}{20c}\delta_{G^{\prime}}(x^{\prime}_{1}).$

Hence

$|z^{\prime}-x^{\prime}_{0}|\leq|z^{\prime}-x^{\prime}_{1}|+|x^{\prime}_{0}-x^{\prime}_{1}|\leq\frac{2q}{21c}\delta_{G^{\prime}}(x^{\prime}_{1})<\frac{q}{10c}\delta_{G^{\prime}}(x^{\prime}_{0}),$

as required.

Next, we need to choose some special points. Let
$x^{\prime}_{3}\in\gamma^{\prime}[x^{\prime},x^{\prime}_{0}]$ be such that

$\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{3}])=\frac{1}{2}\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{0}]),$

and let $x^{\prime}_{2}\in\partial D^{\prime}$ satisfy

$|x^{\prime}_{2}-x^{\prime}_{0}|\leq\frac{3}{2}\delta_{D^{\prime}}(x^{\prime}_{0}).$

Then we assert that

(4.9)
$x^{\prime}_{2},\;x^{\prime}_{3}\in\mathbb{B}(x^{\prime}_{0},q\delta_{G^{\prime}}(x^{\prime}_{0})).$

It follows from (4.8) that we only need to verify the truth of
$x^{\prime}_{2}\in\mathbb{B}(x^{\prime}_{0},q\delta_{G^{\prime}}(x^{\prime}_{0}))$.
Obviously, (4.5) shows that

(4.10)
$|x_{2}^{\prime}-x_{0}^{\prime}|\leq\frac{3}{2}\delta_{D^{\prime}}(x_{0}^{\prime})=\frac{3}{\lambda_{1}}\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{3}])<3\delta_{D^{\prime}}(x_{3}^{\prime}),$

since the choice of $x^{\prime}_{0}$ implies that
$\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{3}])<\lambda_{1}\delta_{D^{\prime}}(x_{3}^{\prime})$.
Also, we have

$\displaystyle k_{D^{\prime}}(x^{\prime}_{0},x^{\prime}_{3})$
$\displaystyle\geq$
$\displaystyle\ell_{k_{D^{\prime}}}(\gamma^{\prime}[x^{\prime}_{0},x^{\prime}_{3}])-\varepsilon\geq\log\left(1+\frac{\ell(\gamma^{\prime}[x^{\prime}_{0},x^{\prime}_{3}])}{\delta_{D^{\prime}}(x^{\prime}_{0})}\right)-1$
$\displaystyle=$ $\displaystyle\log\big{(}1+\frac{\lambda_{1}}{2}\big{)}-1>1,$

whence

$|x_{0}^{\prime}-x_{3}^{\prime}|>\frac{1}{3c}\delta_{D^{\prime}}(x_{0}^{\prime}),$

because otherwise, by Lemma , we get

$k_{D^{\prime}}(x_{0}^{\prime},x_{3}^{\prime})<\frac{1}{3c}\frac{|x_{0}^{\prime}-x_{3}^{\prime}|}{\delta_{D^{\prime}}(x_{0}^{\prime})}\leq\frac{1}{9c^{2}}<1,$

which contradicts with (4.2.1). Hence we deduce from (4.8) and the choice of
$x_{2}^{\prime}$ that

(4.12)
$|x^{\prime}_{2}-x^{\prime}_{0}|\leq\frac{3}{2}\delta_{D^{\prime}}(x^{\prime}_{0})\leq\frac{9c}{2}|x_{0}^{\prime}-x_{3}^{\prime}|\leq\frac{9q}{20}\delta_{G^{\prime}}(x^{\prime}_{0}),$

as needed.

It is (4.9) that allows us to apply Lemma 8(2) to the points $x_{0}^{\prime}$,
$x_{2}^{\prime}$ and $x_{3}^{\prime}$, which shows

$\frac{|x_{2}-x_{0}|}{|x_{3}-x_{0}|}\leq\eta\left(\frac{|x^{\prime}_{2}-x^{\prime}_{0}|}{|x^{\prime}_{3}-x^{\prime}_{0}|}\right),$

and thus, (4.10) implies

(4.13)
$|x^{\prime}_{3}-x^{\prime}_{0}|\leq\frac{1}{\eta^{-1}(\frac{1}{\mu})}|x^{\prime}_{2}-x^{\prime}_{0}|<\frac{3}{\eta^{-1}(\frac{1}{\mu})}\delta_{D^{\prime}}(x^{\prime}_{3}),$

since (4.3) and the choice of $x_{2}^{\prime}$ assure that

$|x_{3}-x_{0}|\leq\mu\delta_{D}(x_{0})\leq\mu|x_{2}-x_{0}|.$

Still, we need an estimate on $|x_{3}-x_{0}|$. Since it follows from Lemma and
(4.2.1) that

$\displaystyle\log\Big{(}1+\frac{|x_{3}-x_{0}|}{\min\\{\delta_{D}(x_{0}),\delta_{D}(x_{3})\\}}\Big{)}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4b_{2}^{2}}k_{D}(x_{0},x_{3})\geq\frac{1}{4b_{2}^{2}M}(k_{D^{\prime}}(x^{\prime}_{0},x^{\prime}_{3})-C)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4b_{2}^{2}M}\Big{(}\log\big{(}1+\frac{\lambda_{1}}{2}\big{)}-1-C\Big{)}$
$\displaystyle>$ $\displaystyle\log(1+\lambda_{2}),$

we know

(4.14)
$|x_{3}-x_{0}|>\lambda_{2}\min\\{\delta_{D}(x_{0}),\delta_{D}(x_{3})\\}>\frac{\lambda_{2}}{1+\mu}\delta_{D}(x_{3}),$

since by (4.3),
$\delta_{D}(x_{3})\leq\delta_{D}(x_{0})+|x_{3}-x_{0}|\leq(1+\mu)\delta_{D}(x_{0}).$

Now, we are ready to get a contradiction to the contrary assumption (4.6). Let
$x_{4}\in\partial D$ be such that

$|x_{4}-x_{3}|\leq 2\delta_{D}(x_{3}).$

Then again by (4.3) and the contrary assumption (4.6), we know that

$\displaystyle|x_{4}-x_{1}|$ $\displaystyle\leq$
$\displaystyle|x_{4}-x_{3}|+|x_{3}-x_{1}|\leq
2\delta_{D}(x_{3})+|x_{3}-x_{1}|\leq 2\delta_{D}(x_{1})+3|x_{3}-x_{1}|$
$\displaystyle\leq$
$\displaystyle(2+3\mu)\delta_{D}(x_{1})<\frac{2+3\mu}{\lambda_{2}}\delta_{G}(x_{1})$
$\displaystyle<$ $\displaystyle\frac{q}{2}\delta_{G}(x_{1}),$

which, together with (4.7), implies

$x_{0},\;x_{3},\;x_{4}\in\mathbb{B}(x_{1},\frac{q}{2}\delta_{G}(x_{1})).$

Apply Lemma 8(2) to the points $x_{0},$ $x_{3},$ and $x_{4}$. Then we see from
(4.13), (4.14) and the choice of $x_{4}$ that

$\frac{1}{3}\eta^{-1}\big{(}\frac{1}{\mu}\big{)}\leq\frac{|x^{\prime}_{4}-x^{\prime}_{3}|}{|x^{\prime}_{3}-x^{\prime}_{0}|}\leq\eta\left(\frac{|x_{4}-x_{3}|}{|x_{3}-x_{0}|}\right)<\eta\left(\frac{2(1+\mu)}{\lambda_{2}}\right),$

which is the desired contradiction since
$\lambda_{2}\geq\frac{2(1+\mu)}{\eta^{-1}\big{(}\frac{1}{3}\eta^{-1}\big{(}\frac{1}{\mu}\big{)}\big{)}}$.
Hence Claim 4.1 holds.

If $(2)$ happens, then we let $y^{\prime}_{0}$ be the first point in
$\gamma^{\prime}[y^{\prime},z^{\prime}_{0}]$ from $y^{\prime}$ to
$z^{\prime}_{0}$ such that

$\gamma^{\prime}[y^{\prime},y^{\prime}_{0}]=\lambda_{1}\delta_{D^{\prime}}(y^{\prime}_{0}),$

and the similar reasoning as in the proof of Claim 4.1 implies

######  Claim 4.2.

For every $z\in\gamma[y_{0},z_{0}]$, we have
$\delta_{G}(z)\leq\lambda_{2}\delta_{D}(z)$.

In order to prove that $\gamma^{\prime}$ satisfies the cigar condition with
constant $2\lambda_{1}^{2}$, we only need to consider the case where both (1)
and (2) happen because the proofs for other cases are similar, and in fact,
the corresponding discussions are simpler. First, we partition the part
$\gamma^{\prime}[x_{0}^{\prime},y_{0}^{\prime}]$ of $\gamma^{\prime}$ as
follows. Let $u^{\prime}_{0}\in\gamma^{\prime}[x^{\prime}_{0},y^{\prime}_{0}]$
be such that

$\delta_{D^{\prime}}(u^{\prime}_{0})=\max_{p^{\prime}\in\gamma^{\prime}[x^{\prime}_{0},y^{\prime}_{0}]}\delta_{D^{\prime}}(p^{\prime}).$

Obviously, there exists a unique integer $k\geq 0$ with

$2^{k}\delta_{D^{\prime}}(x^{\prime}_{0})\leq\delta_{D^{\prime}}(u^{\prime}_{0})<2^{k+1}\delta_{D^{\prime}}(x^{\prime}_{0}).$

Let $v^{\prime}_{0}=x^{\prime}_{0}$. If $k=0$, then we let
$v^{\prime}_{1}=u^{\prime}_{0}$. If $k>1$, then for each $i\in\\{1,$ $\ldots,$
$k\\}$, we let $v^{\prime}_{i}$ be the first point in
$\gamma^{\prime}[x^{\prime}_{0},u^{\prime}_{0}]$ from $v^{\prime}_{i-1}$ to
$u^{\prime}_{0}$ such that

$\delta_{D^{\prime}}(v^{\prime}_{i})=2^{i}\delta_{D^{\prime}}(x^{\prime}_{0}),$

and let $v^{\prime}_{k+1}=u^{\prime}_{0}$. It is possible that
$v^{\prime}_{k+1}=v^{\prime}_{k}$. This possibility happens if and only if
$\delta_{D^{\prime}}(u^{\prime}_{0})=2^{k}\delta_{D^{\prime}}(x^{\prime}_{0})$.

As for this partition, we have the following assertion.

######  Claim 4.3.

For each $i\in\\{0,$ $\ldots,$ $k\\}$, we have

  1. (1)

$\ell(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])\leq\lambda^{4}_{2}\delta_{D^{\prime}}(v^{\prime}_{i}),$
and

  2. (2)

for $z^{\prime}\in\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}]$,

$\delta_{D^{\prime}}(v^{\prime}_{i})<(1+\lambda_{2}^{4})^{20b_{1}^{2}b_{2}^{2}M^{2}\lambda_{2}}\delta_{D^{\prime}}(z^{\prime}).$

We first prove (1). It follows from Lemmas and 8(1) that

$\displaystyle\ell_{k_{D^{\prime}}}(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])$
$\displaystyle\leq$ $\displaystyle
k_{D^{\prime}}(v^{\prime}_{i},v^{\prime}_{i+1})+\varepsilon\leq
Mk_{D}(v_{i},v_{i+1})+C+1$ $\displaystyle\leq$ $\displaystyle
4b_{2}^{2}M\log\left(1+\frac{|v_{i}-v_{i+1}|}{\min\\{\delta_{D}(v_{i}),\delta_{D}(v_{i+1})\\}}\right)+C+1,$

and thus, Claims 4.1 and 4.2, together with Lemma 8(1), lead to

$\displaystyle\ell_{k_{D^{\prime}}}(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])$
$\displaystyle\leq$ $\displaystyle
4b_{2}^{2}M\log\left(1+\frac{\lambda_{2}|v_{i}-v_{i+1}|}{\min\\{\delta_{G}(v_{i}),\delta_{G}(v_{i+1})\\}}\right)+C+1$
$\displaystyle\leq$ $\displaystyle
4b_{2}^{2}M\lambda_{2}k_{G}(v_{i},v_{i+1})+C+1$ $\displaystyle\leq$
$\displaystyle
4b_{2}^{2}M^{2}\lambda_{2}k_{G^{\prime}}(v^{\prime}_{i},v^{\prime}_{i+1})+4b_{2}^{2}CM\lambda_{2}+C+1,$

and finally, with the aid of Lemma , we obtain

$\displaystyle\ell_{k_{D^{\prime}}}(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])$
$\displaystyle\leq$ $\displaystyle
16b_{1}^{2}b_{2}^{2}M^{2}\lambda_{2}\log\left(1+\frac{|v^{\prime}_{i}-v^{\prime}_{i+1}|}{\min\\{\delta_{G^{\prime}}(v^{\prime}_{i}),\delta_{G^{\prime}}(v^{\prime}_{i+1})\\}}\right)$
$\displaystyle+4b_{2}^{2}CM\lambda_{2}+C+1$ $\displaystyle<$ $\displaystyle
16b_{1}^{2}b_{2}^{2}M^{2}\lambda_{2}\log\left(1+\frac{\ell(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])}{\delta_{D^{\prime}}(v^{\prime}_{i})}\right)+4b_{2}^{2}CM\lambda_{2}$
$\displaystyle+C+1.$

Meanwhile, the choice of $v^{\prime}_{i}$ gives

$\ell_{k_{D^{\prime}}}(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])=\int_{\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}]}\frac{|dz|}{\delta_{D^{\prime}}(z)}\geq\frac{\ell(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])}{2\delta_{D^{\prime}}(v^{\prime}_{i})},$

and so,

$\ell(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])\leq\lambda^{4}_{2}\delta_{D^{\prime}}(v^{\prime}_{i}).$

Thus Claim 4.3(1) holds.

Moreover, by (4.2.1) and Claim 4.3(1), we see that for all
$z^{\prime}\in\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}]$,

$\displaystyle\log\frac{\delta_{D^{\prime}}(v^{\prime}_{i})}{\delta_{D^{\prime}}(z^{\prime})}$
$\displaystyle\leq$ $\displaystyle
k_{D^{\prime}}(v^{\prime}_{i},z^{\prime})\leq\ell_{k_{D^{\prime}}}(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])$
$\displaystyle\leq$ $\displaystyle
16b_{1}^{2}b_{2}^{2}M^{2}\lambda_{2}\log(1+\lambda_{2}^{4})+4b_{2}^{2}CM\lambda_{2}+C+1$
$\displaystyle\leq$ $\displaystyle
20b_{1}^{2}b_{2}^{2}M^{2}\lambda_{2}\log(1+\lambda_{2}^{4}).$

Hence

$\delta_{D^{\prime}}(v^{\prime}_{i})<(1+\lambda_{2}^{4})^{20b_{1}^{2}b_{2}^{2}M^{2}\lambda_{2}}\delta_{D^{\prime}}(z^{\prime}),$

which shows that Claim 4.3(2) is true too, and thus the claim is proved.

Now, we are ready to verify that $\gamma^{\prime}$ satisfies the
$2\lambda_{1}^{2}$-cigar condition, that is, for every
$z^{\prime}\in\gamma^{\prime}[x^{\prime},y^{\prime}]$,

(4.16)
$\min\\{\ell(\gamma^{\prime}[x^{\prime},z^{\prime}]),\ell(\gamma^{\prime}[z^{\prime},y^{\prime}])\\}\leq
2\lambda_{1}^{2}\delta_{D^{\prime}}(z^{\prime}).$

Obviously, we only need to consider the case
$z^{\prime}\in\gamma^{\prime}[x^{\prime},u^{\prime}_{0}]$ since the proof for
the case $z^{\prime}\in\gamma^{\prime}[y^{\prime},u^{\prime}_{0}]$ is similar.

If $z^{\prime}\in\gamma^{\prime}[x^{\prime},x^{\prime}_{0}]$, then (4.16)
immediately follows from the choice of $x^{\prime}_{0}$.

If $z^{\prime}\in\gamma^{\prime}[x^{\prime}_{0},u^{\prime}_{0}]$, then there
must exist a $j\in\\{0,\ldots,k\\}$ such that

$z^{\prime}\in\gamma^{\prime}[v^{\prime}_{j},v^{\prime}_{j+1}],$

and so by (4.5) and Claim 4.3, we have the following:

$\displaystyle\ell(\gamma^{\prime}[x^{\prime},z^{\prime}])$ $\displaystyle=$
$\displaystyle\ell(\gamma^{\prime}[x^{\prime},x^{\prime}_{0}])+\ell(\gamma^{\prime}[x^{\prime}_{0},z^{\prime}])\leq\lambda_{1}\delta_{D^{\prime}}(x^{\prime}_{0})+\sum_{i=0}^{j}\ell(\gamma^{\prime}[v^{\prime}_{i},v^{\prime}_{i+1}])$
$\displaystyle\leq$
$\displaystyle\lambda_{1}\delta_{D^{\prime}}(v^{\prime}_{j})+\lambda_{2}^{4}\sum_{i=0}^{j}\delta_{D^{\prime}}(v^{\prime}_{i})\leq(\lambda_{1}+2\lambda_{2}^{4})\delta_{D^{\prime}}(v^{\prime}_{j})$
$\displaystyle<$ $\displaystyle
2\lambda_{1}^{2}\delta_{D^{\prime}}(z^{\prime}),$

which shows that (4.16) is also true in this case. Hence $\gamma^{\prime}$
satisfies the $2\lambda_{1}^{2}$-cigar condition. ∎

####  4.2.2. The verification of the turning condition

We only need to prove that

(4.17) $\ell(\gamma^{\prime})\leq\lambda_{0}|x^{\prime}-y^{\prime}|,$

where

$\lambda_{0}=\max\Big{\\{}24c\lambda_{1}^{2}e^{1+C}e^{6Mb_{2}^{2}[1+\mu\eta(8\lambda_{1}^{2}+\frac{1}{3c})]},48c\lambda_{1}^{2}e^{\rho}\Big{\\}}$

and

$\rho=\frac{96(c+1)(b_{1}b_{2}M)^{2}}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}(\frac{q}{12\lambda_{1}^{2}})}\log\Big{(}1+4c\lambda_{1}^{2}+\frac{1}{6c}\Big{)}+\frac{24(c+1)MCb_{2}^{2}}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}(\frac{q}{12\lambda_{1}^{2}})}+C+1.$

We prove (4.17) by contradiction. Suppose that

(4.18) $\ell(\gamma^{\prime})>\lambda_{0}|x^{\prime}-y^{\prime}|.$

Then Lemma 2 implies

$|x^{\prime}-y^{\prime}|>\frac{1}{3c}\max\\{\delta_{D^{\prime}}(x^{\prime}),\delta_{D^{\prime}}(y^{\prime})\\},$

and also, we know that there exist $w^{\prime}_{1}$,
$w^{\prime}_{2}\in\gamma^{\prime}$ such that $x^{\prime}$, $w^{\prime}_{1}$,
$w^{\prime}_{2}$ and $y^{\prime}$ are successive pints in $\gamma^{\prime}$,
and

(4.19)
$\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{1}])=\ell(\gamma^{\prime}[w^{\prime}_{2},y^{\prime}])=12c\lambda_{1}^{2}|x^{\prime}-y^{\prime}|.$

As for this partition of $\gamma^{\prime}$, we prove several claims.

######  Claim 4.4.

$|x^{\prime}-w^{\prime}_{1}|\geq\frac{1}{2}\delta_{D^{\prime}}(w^{\prime}_{1})\;\;{\rm
and}\;\;|y^{\prime}-w^{\prime}_{2}|\geq\frac{1}{2}\delta_{D^{\prime}}(w^{\prime}_{2}).$

The proof of this claim easily follows from a similar argument as in the proof
of Claim 3.4 with the substitution $A$ by $2\lambda_{1}^{2}$.

######  Claim 4.5.

$6c|x^{\prime}-y^{\prime}|\leq\min\\{\delta_{D^{\prime}}(w^{\prime}_{1}),\;\delta_{D^{\prime}}(w^{\prime}_{2})\\}\leq\max\\{\delta_{D^{\prime}}(w^{\prime}_{1}),\;\delta_{D^{\prime}}(w^{\prime}_{2})\\}\leq
24c\lambda_{1}^{2}|x^{\prime}-y^{\prime}|.$

It is equivalent to show that
$6c|x^{\prime}-y^{\prime}|\leq\delta_{D^{\prime}}(w^{\prime}_{i})\leq
24c\lambda_{1}^{2}|x^{\prime}-y^{\prime}|$ for $i\in\\{1,\;2\\}$, which easily
follows from Claim 4.4, (4.16) and (4.19).

######  Claim 4.6.

$|w_{1}-w_{2}|>\mu\eta\big{(}8\lambda_{1}^{2}+\frac{1}{3c}\big{)}\min\\{\delta_{D}(w_{1}),\delta_{D}(w_{2})\\}.$

For convenience, we may assume that

$\min\\{\delta_{D}(w_{1}),\delta_{D}(w_{2})\\}=\delta_{D}(w_{1}).$

Since $D$ is $b_{2}$-uniform, we see from Lemma that

$\log\left(1+\frac{|w_{1}-w_{2}|}{\delta_{D}(w_{1})}\right)\geq\frac{1}{4b_{2}^{2}}k_{D}(w_{1},w_{2})\geq\frac{1}{4Mb_{2}^{2}}(k_{D^{\prime}}(w^{\prime}_{1},w^{\prime}_{2})-C),$

and thus, the assumption “$\gamma^{\prime}$ being an $\varepsilon$-short arc”
implies

$\displaystyle\log\left(1+\frac{|w_{1}-w_{2}|}{\delta_{D}(w_{1})}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4Mb_{2}^{2}}(\ell_{k_{D^{\prime}}}(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])-\varepsilon-C)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4Mb_{2}^{2}}\left(\log\left(1+\frac{\ell(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])}{\delta_{D^{\prime}}(w^{\prime}_{1})}\right)-1-C\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4Mb_{2}^{2}}\left(\log\left(1+\frac{\lambda_{0}-24c\lambda_{1}^{2}}{24c\lambda_{1}^{2}}\right)-1-C\right)$
$\displaystyle=$ $\displaystyle\mu_{1},$

where the last inequality holds because of Claim 4.5 and the following chain
of inequalities which are from (4.18) and (4.19):

(4.20)
$\ell(\gamma^{\prime}[w_{1}^{\prime},w_{2}^{\prime}])=\ell(\gamma^{\prime})-\ell(\gamma^{\prime}[x^{\prime},w_{1}^{\prime}])-\ell(\gamma^{\prime}[y^{\prime},w_{2}^{\prime}])>(\lambda_{0}-24c\lambda_{1}^{2})|x^{\prime}-y^{\prime}|.$

Hence

$|w_{1}-w_{2}|\geq(e^{\mu_{1}}-1)\delta_{D}(w_{1})>\mu\eta\big{(}8\lambda_{1}^{2}+\frac{1}{3c}\big{)}\delta_{D}(w_{1}),$

as needed.

With the aid of (4.3), the following is a direct consequence of Claim 4.6.

######  Claim 4.7.

$|x-w_{1}|\leq\mu\delta_{D}(w_{1}).$

For points in $\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}]$, we have the
following comparison result.

######  Claim 4.8.

For all $w^{\prime}\in\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}]$,
$\delta_{G^{\prime}}(w^{\prime})\leq\frac{12\lambda_{1}^{2}}{q}\delta_{D^{\prime}}(w^{\prime})$.

Suppose that there exists some point
$w^{\prime}_{3}\in\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}]$ with

(4.21)
$\delta_{G^{\prime}}(w^{\prime}_{3})>\frac{12\lambda_{1}^{2}}{q}\delta_{D^{\prime}}(w^{\prime}_{3}).$

Without loss of generality, we may assume that

$\min\\{\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{3}]),\ell(\gamma^{\prime}[y^{\prime},w^{\prime}_{3}])\\}=\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{3}]),$

since the proof for the other case is similar.

By (4.16) and (4.21), for all
$w^{\prime}\in\gamma^{\prime}[x^{\prime},w^{\prime}_{3}]$, we have

$\ell(\gamma^{\prime}[w^{\prime},w^{\prime}_{3}])\leq
2\lambda_{1}^{2}\delta_{D^{\prime}}(w^{\prime}_{3})<\frac{q}{6}\delta_{G^{\prime}}(w_{3}^{\prime}),$

and so, (4.19) gives

$\displaystyle|w_{2}^{\prime}-w^{\prime}_{3}|$ $\displaystyle\leq$
$\displaystyle|w_{2}^{\prime}-y^{\prime}|+|y^{\prime}-x^{\prime}|+|x^{\prime}-w^{\prime}_{3}|$
$\displaystyle\leq$
$\displaystyle\ell(\gamma^{\prime}[y^{\prime},w_{2}^{\prime}])+\frac{1}{12c\lambda_{1}^{2}}\ell(\gamma^{\prime}[x^{\prime},w_{1}^{\prime}])+\ell(\gamma^{\prime}[x^{\prime},w_{3}^{\prime}])$
$\displaystyle\leq$
$\displaystyle\big{(}2+\frac{1}{12c\lambda_{1}^{2}}\big{)}\ell(\gamma^{\prime}[x^{\prime},w^{\prime}_{3}])<\frac{q}{2}\delta_{G^{\prime}}(w_{3}^{\prime}).$

Then we know that

$x^{\prime},\;w^{\prime}_{1},\;w^{\prime}_{2}\in\mathbb{B}(w^{\prime}_{3},\frac{q}{2}\delta_{G^{\prime}}(w^{\prime}_{3})).$

By applying Lemma 8(2) to the points $x^{\prime}$, $w^{\prime}_{1}$ and
$w^{\prime}_{2}$, together with (4.2.2) and Claims 4.4 $\sim$ 4.7, we know
that

$\eta\big{(}8\lambda_{1}^{2}+\frac{1}{3c}\big{)}<\frac{|w_{2}-w_{1}|}{|x-w_{1}|}\leq\eta\left(\frac{|w^{\prime}_{1}-w^{\prime}_{2}|}{|x^{\prime}-w^{\prime}_{1}|}\right)\leq\eta\big{(}8\lambda_{1}^{2}+\frac{1}{3c}\big{)},$

since by (4.19),

$|w^{\prime}_{1}-w^{\prime}_{2}|\leq|w^{\prime}_{2}-y^{\prime}|+|y^{\prime}-x^{\prime}|+|x^{\prime}-w^{\prime}_{1}|\leq(1+24c\lambda_{1}^{2})|x^{\prime}-y^{\prime}|.$

This is the desired contradiction, from which the claim follows.

The next result is an analogue of Claim 4.8 for points in
$\gamma[w_{1},w_{2}]$.

######  Claim 4.9.

For all $w\in\gamma[w_{1},w_{2}]$, we have

$\delta_{G}(w)\leq\frac{6(c+1)}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}\delta_{D}(w).$

Suppose that there exists some point $u\in\gamma[w_{1},w_{2}]$ such that

$\delta_{G}(u)>\frac{6(c+1)}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}\delta_{D}(u).$

Let

$u_{1}\in\mathbb{S}\Big{(}u,\frac{q\varphi^{-1}(\log\frac{3}{2})}{3(c+1)}\delta_{G}(u)\Big{)}.$

Then Lemma 3 guarantees that $u_{1}\in G$, and thus, Lemma implies

$k_{G^{\prime}}(u_{1}^{\prime},u^{\prime})\leq\varphi(k_{G}(u_{1},u))\leq\varphi\Big{(}3c\frac{|u_{1}-u|}{\delta_{G}(u)}\Big{)}<\log\frac{3}{2},$

which leads to

(4.22)
$|u_{1}^{\prime}-u^{\prime}|\leq\frac{1}{2}\delta_{G^{\prime}}(u^{\prime}).$

Let $u_{2}\in\partial D$ be such that

$|u_{2}-u|\leq 2\delta_{D}(u).$

Thus

(4.23) $|u_{2}-u|\leq
2\delta_{D}(u)\leq\frac{q\varphi^{-1}(\log\frac{3}{2})}{3(c+1)}\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}\delta_{G}(u)<\frac{q}{3(c+1)}\delta_{G}(u).$

Hence, the choice of $u_{1}$ implies

$u_{1},\;u_{2}\in\mathbb{B}(u,\frac{q}{2}\delta_{G}(u))\cap G.$

Apply Lemma 8(2) to the points $u$, $u_{1}$ and $u_{2}$. Then (4.22) and
(4.23) lead to

$\frac{q}{6\lambda_{1}^{2}}<\frac{|u^{\prime}_{2}-u^{\prime}|}{|u^{\prime}_{1}-u^{\prime}|}\leq\eta\left(\frac{|u_{2}-u|}{|u_{1}-u|}\right)<\frac{q}{12\lambda_{1}^{2}},$

since Claim 4.8 leads to

$|u^{\prime}_{2}-u^{\prime}|\geq\delta_{D^{\prime}}(u^{\prime})\geq\frac{q}{12\lambda_{1}^{2}}\delta_{G^{\prime}}(u^{\prime}).$

This obvious contradiction completes the proof.

Now, we are ready to get a contradiction to the contrary assumption (4.18). By
Claim 4.5 and (4.20), we have

$\log\left(1+\frac{\ell(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])}{\min\\{\delta_{D^{\prime}}(w^{\prime}_{1}),\delta_{D^{\prime}}(w^{\prime}_{2})\\}}\right)\geq\log\left(1+\frac{\lambda_{0}-24c\lambda_{1}^{2}}{24c\lambda_{1}^{2}}\right).$

Also, Lemma implies

$\displaystyle\log\left(1+\frac{\ell(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])}{\min\\{\delta_{D^{\prime}}(w^{\prime}_{1}),\delta_{D^{\prime}}(w^{\prime}_{2})\\}}\right)$
$\displaystyle\leq$
$\displaystyle\ell_{k_{D^{\prime}}}(\gamma^{\prime}[w^{\prime}_{1},w^{\prime}_{2}])\leq
k_{D^{\prime}}(w^{\prime}_{1},w^{\prime}_{2})+\varepsilon$ $\displaystyle\leq$
$\displaystyle Mk_{D}(w_{1},w_{2})+C+1$ $\displaystyle\leq$ $\displaystyle
4b_{2}^{2}M\log\left(1+\frac{|w_{1}-w_{2}|}{\min\\{\delta_{D}(w_{1}),\delta_{D}(w_{2})\\}}\right)+C+1.$

Furthermore, Claim 4.9 gives

$\min\\{\delta_{D}(w_{1}),\delta_{D}(w_{2})\\}\geq\frac{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}{6(c+1)}\min\\{\delta_{G}(w_{1}),\delta_{G}(w_{2})\\},$

and Lemma 8(1) leads to

$k_{G}(w_{1},w_{2})\leq Mk_{G^{\prime}}(w^{\prime}_{1},w^{\prime}_{2})+C.$

Consequently, it follows from Lemma that

$\displaystyle\log\left(1+\frac{\lambda_{0}-24c\lambda_{1}^{2}}{24c\lambda_{1}^{2}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{24b_{2}^{2}(c+1)M^{2}}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}k_{G^{\prime}}(w^{\prime}_{1},w^{\prime}_{2})$
$\displaystyle+$
$\displaystyle\frac{24b_{2}^{2}(c+1)CM}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}+C+1$
$\displaystyle\leq$
$\displaystyle\frac{96b_{1}^{2}b_{2}^{2}(c+1)CM^{2}}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}\log\left(1+\frac{|w^{\prime}_{1}-w^{\prime}_{2}|}{\min\\{\delta_{G^{\prime}}(w^{\prime}_{1}),\delta_{G^{\prime}}(w^{\prime}_{2})\\}}\right)$
$\displaystyle+\frac{24b_{2}^{2}(c+1)CM}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}+C+1.$

Since (4.19) and Claim 4.5 lead to

$\displaystyle|w^{\prime}_{1}-w^{\prime}_{2}|$ $\displaystyle\leq$
$\displaystyle|w^{\prime}_{1}-x^{\prime}|+|x^{\prime}-y^{\prime}|+|y^{\prime}-w^{\prime}_{2}|\leq(24c\lambda_{1}^{2}+1)|x^{\prime}-y^{\prime}|$
$\displaystyle\leq$
$\displaystyle\big{(}4\lambda_{1}^{2}+\frac{1}{6c}\big{)}\min\\{\delta_{G^{\prime}}(w^{\prime}_{1}),\delta_{G^{\prime}}(w^{\prime}_{2})\\}.$

Finally, we see that

$\displaystyle\log\left(1+\frac{\lambda_{0}-24c\lambda_{1}^{2}}{24c\lambda_{1}^{2}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{96b_{1}^{2}b_{2}^{2}(c+1)CM^{2}}{q\eta^{-1}\big{(}\frac{q\varphi^{-1}(\log\frac{3}{2})}{12\lambda_{1}^{2}}\big{)}}\log\Big{(}1+4\lambda_{1}^{2}+\frac{1}{6c}\Big{)}$
$\displaystyle+\frac{24b_{2}^{2}(c+1)CM}{q\varphi^{-1}(\log\frac{3}{2})\eta^{-1}\big{(}\frac{q}{12\lambda_{1}^{2}}\big{)}}+C+1$
$\displaystyle=$ $\displaystyle\rho,$

which is the desired contradiction since $\lambda_{0}\geq
48c\lambda_{1}^{2}e^{\rho}$. Hence Theorem 2 holds.∎

##  5\. The proof of Theorem 3

We start with two notations. Let $f:X\to Y$ be a homeomorphism between two
metric spaces, and let $x$ be a non-isolated point of $X$. We write

$L(x,f)=\limsup_{y\to
x}\frac{|y^{\prime}-x^{\prime}|}{|y-x|}\;\;\mbox{and}\;\;l(x,f)=\liminf_{y\to
x}\frac{|y^{\prime}-x^{\prime}|}{|y-x|}.$

Suppose $G$ denotes a proper subdomain in $X$. If $|\cdot|=k_{G}(\cdot)$, then
we denote $L(x,f)$ and $l(x,f)$ by $L_{k_{G}}(x,f)$ and $l_{k_{G}}(x,f)$,
respectively.

Now, we are going to show three lemmas. The first lemma is about the
comparison of the quantities $L(x,f)$ and $L_{k_{G}}(x,f)$ (resp. $l(x,f)$ and
$l_{k_{G}}(x,f)$).

######  Lemma 9.

Suppose that $X$ is $c$-quasiconvex and $Y$ is $c^{\prime}$-quasiconvex, and
that $G\subsetneq X$ and $G^{\prime}\subsetneq Y$ are domains. If $f:G\to
G^{\prime}$ is continuous, then

$\frac{L(x,f)\delta_{G}(x)}{6c\delta_{G^{\prime}}(x^{\prime})}\leq
L_{k_{G}}(x,f)\leq
6c^{\prime}\frac{L(x,f)\delta_{G}(x)}{\delta_{G^{\prime}}(x^{\prime})}$

and

$\frac{l(x,f)\delta_{G}(x)}{6c\delta_{G^{\prime}}(x^{\prime})}\leq
l_{k_{G}}(x,f)\leq
6c^{\prime}\frac{l(x,f)\delta_{G}(x)}{\delta_{G^{\prime}}(x^{\prime})}.$

Proof. By symmetry, it suffices to prove the first chain of inequalities in
the lemma. On the one hand, by Lemma , we have

$\displaystyle L_{k_{G}}(x,f)$ $\displaystyle=$ $\displaystyle\limsup_{y\to
x}\frac{k_{G^{\prime}}(x^{\prime},y^{\prime})}{k_{G}(x,y)}=\limsup_{y\to
x}\left(\frac{k_{G^{\prime}}(x^{\prime},y^{\prime})}{|x^{\prime}-y^{\prime}|}\frac{|x^{\prime}-y^{\prime}|}{|x-y|}\frac{|x-y|}{k_{G}(x,y)}\right)$
$\displaystyle\leq$ $\displaystyle
L(x,f)\frac{6c^{\prime}\delta_{G}(x)}{\delta_{G^{\prime}}(x^{\prime})}.$

On the other hand, again by Lemma , we know

$\displaystyle L(x,f)$ $\displaystyle=$ $\displaystyle\limsup_{y\to
x}\frac{|x^{\prime}-y^{\prime}|}{|x-y|}$ $\displaystyle=$
$\displaystyle\limsup_{y\to
x}\left(\frac{|x^{\prime}-y^{\prime}|}{k_{G^{\prime}}(x^{\prime},y^{\prime})}\frac{k_{G^{\prime}}(x^{\prime},y^{\prime})}{k_{G}(x,y)}\frac{k_{G}(x,y)}{|x-y|}\right)$
$\displaystyle\leq$ $\displaystyle
L_{k_{G}}(x,f)\frac{6c\delta_{G^{\prime}}(x^{\prime})}{\delta_{G}(x)}.$

Hence the proof is complete. ∎

The next lemma is a characterization for a homeomorphism from $X$ to $Y$ to be
$M$-QH in terms of $l_{k_{G}}(x,f)$ and $L_{k_{G}}(x,f)$.

######  Lemma 10.

Suppose $f:G\to G^{\prime}$ is a homeomorphism, where $G$ denotes a proper
subdomain of $X$. Then $f$ is $M$-QH if and only if

$\frac{1}{M}\leq l_{k_{G}}(x,f)\leq L_{k_{G}}(x,f)\leq M.$

Proof. Since the spaces $(G,k_{G})$ and $(G^{\prime},k_{G^{\prime}})$ are
$\tau$-quasiconvex for all $\tau>1$ (see [23, Lemma 2.5]), we easily know from
[37, Lemma 5.5] that the lemma holds. ∎

We are ready to prove the main lemma in this section.

######  Lemma 11.

Suppose that $X$ is $c$-quasiconvex and $Y$ is $c^{\prime}$-quasiconvex, and
that $G\subsetneq X$ and $G^{\prime}\subsetneq Y$ are domains. If $f:G\to
G^{\prime}$ is $M$-QH, then $f$ is fully $M^{\prime}$-QH with
$M^{\prime}=216(cc^{\prime}M)^{2}$. That is, $f$ is $\varphi$-FQC with
$\varphi(t)=M^{\prime}t$.

Proof. Let $D$ be an arbitrary domain in $G$. Fix a point $x$ in $D$. By
symmetry and Lemma 10, it suffices to show that

(5.1) $L_{k_{D}}(x,f)\leq M^{\prime}.$

Since $f$ is $M$-QH, Lemma 10 gives $L_{k_{G}}(x,f)\leq M$. Hence, to prove
(5.1), by Lemma 9, it suffices to show that

(5.2) $\frac{\delta_{D}(x)}{\delta_{D^{\prime}}(x^{\prime})}\leq
6cc^{\prime}M\frac{\delta_{G}(x)}{\delta_{G^{\prime}}(x^{\prime})}.$

If
$\delta_{D^{\prime}}(x^{\prime})\geq\frac{1}{6cc^{\prime}M}\delta_{G^{\prime}}(x^{\prime})$,
then (5.2) is obvious. So we assume in the following that

$\delta_{D^{\prime}}(x^{\prime})<\frac{1}{6cc^{\prime}M}\delta_{G^{\prime}}(x^{\prime}).$

Let

$0<\epsilon<\frac{1}{6cc^{\prime}M}\delta_{G^{\prime}}(x^{\prime})-\delta_{D^{\prime}}(x^{\prime}),$

and let $y^{\prime}\in\partial D^{\prime}$ be such that

(5.3)
$|y^{\prime}-x^{\prime}|<\delta_{D^{\prime}}(x^{\prime})+\epsilon<\frac{1}{6cc^{\prime}M}\delta_{G^{\prime}}(x^{\prime}).$

Then by Lemma 3, we know $y^{\prime}\in G^{\prime}$, and thus, Lemma leads to

$k_{G^{\prime}}(x^{\prime},y^{\prime})\leq
3c^{\prime}\frac{|x^{\prime}-y^{\prime}|}{\delta_{G^{\prime}}(x^{\prime})}\leq\frac{1}{2cM},$

which shows that

$k_{G}(x,y)\leq Mk_{G^{\prime}}(x^{\prime},y^{\prime})\leq\frac{1}{2c}.$

Hence again by Lemma , together with (5.3), we have

$\displaystyle\delta_{D}(x)$ $\displaystyle\leq$ $\displaystyle|x-y|\leq
2k_{G}(x,y)\delta_{G}(x)\leq
2Mk_{G^{\prime}}(x^{\prime},y^{\prime})\delta_{G}(x)\leq
6c^{\prime}M\frac{|x^{\prime}-y^{\prime}|}{\delta_{G^{\prime}}(x^{\prime})}\delta_{G}(x)$
$\displaystyle\leq$ $\displaystyle
6c^{\prime}M\frac{\delta_{D^{\prime}}(x^{\prime})+\epsilon}{\delta_{G^{\prime}}(x^{\prime})}\delta_{G}(x).$

Letting $\epsilon\to 0$ gives (5.2). Hence the proof of Lemma 11 is complete.
∎

The proof of Theorem 3. Obviously, the proof of Theorem 3 easily follows from
Theorem 2 and Lemma 11.

## References

  * [1] L. V. Ahlfors, Quasiconformal reflections, Acta Math., 109 (1963), 291–301.
  * [2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Dimension-free quasiconformal distortion in $n$-space. Trans. Amer. Math. Soc., 297 (1986), 687–706.
  * [3] A. F. Beardon, The Apollonian metric of a domain in ${\mathbb{R}}^{n}$, In: Quasiconformal mappings and analysis, Springer-Verlag, 1998, 91–108.
  * [4] A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings,  Acta Math., 96 (1956), 125–142.
  * [5] M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic domains, Asterisque 270 (2001), 1–99.
  * [6] St. M. Buckley and D. A. Herron, Uniform spaces and weak slice spaces, Conform. Geom. Dyn., 11 (2007), 191 C206 (electronic).
  * [7] St. M. Buckley, D. A. Herron and X. Xie, Metric space inversions, quasihyperbolic distance, and uniform spaces,  Indiana Univ. Math. J., 57 (2008), 837–890.
  * [8] St. M. Buckley, P. Koskela and G. Lu, Boman equals John. In: XVI Rolf Nevanlinna Colloquium: proceedings of the international conference held in Joensuu, Finland, August 1-5, 1995. W. de Gruyter, Berlin, New York, pp. 91-99.
  * [9] J. L. Fernández, J. Heinonen and O. Martio, Quasilines and conformal mappings, J. Analyse Math., 52 (1989), 117–132.
  * [10] F. W. Gehring, Uniform domains and the ubiquitous quasidisks, Jahresber. Deutsch. Math. Verein, 89 (1987), 88–103.
  * [11] F. W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings. J. Analyse Math., 45 (1985), 181–206.
  * [12] F. W. Gehring and B. G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Analyse Math., 36 (1979), 50–74.
  * [13] F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math., 30 (1976), 172–199.
  * [14] P. Hästö, The Apollonian metric: uniformity and quasiconvexity, Ann. Acad. Sci. Fenn. Ser. Math., 28 (2003), 385–414.
  * [15] J. Heinonen, Quasiconformal mappings onto John domains, Rev. Mat. Iberoamericana, 5 (1989), 97-123.
  * [16] J. Heinonen, Lectures on analysis on metric spaces, Springer, 2001.
  * [17] J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math., 120 (1995), 61–79.
  * [18] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry,  Acta Math., 181 (1998), 1–61.
  * [19] D. Herron, N. Shanmugalingam and X. Xie, Uniformity from Gromov hyperbolicity, Illinois J. Math., 52 (2008), 1065–1109.
  * [20] M. Huang and Y. Li, Decomposition properties of John domains in real normed vector spaces, J. Math. Anal. Appl., 388 (2012), 191-197.
  * [21] M. Huang, Y. Li, S. Ponnusamy and X. Wang, The quasiconformal subinvariance property of John domians in ${\mathbb{R}}^{n}$ and its applications, To appear in Math. Ann..
  * [22] M. Huang, S. Ponnusamy, A. Rasila and X. Wang, On quasisymmetry of quasiconformal mappings and its applications, arXiv: 1207.4360v2 [math. CV], 2013.
  * [23] M. Huang, A. Rasila, X. Wang and Q. Zhou, The characterizations of free quasiconformal mappings on metric spaces, arXiv: 1409.5631 [math. CV], 2014.
  * [24] M. Huang, M. Vuorinen and X. Wang, On the subinvariance of uniform domains in Banach spaces, J. Math. Anal. Appl., 407 (2013), 527-540.
  * [25] X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, To appear in Trans. Amer. Math. Soc..
  * [26] F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.
  * [27] P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J., 29 (1980), 41–66.
  * [28] Y. Li and X. Wang, Unions of John domains and uniform domains in real normed vector spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., 35 (2010), 627–632.
  * [29] O. Martio, Definitions of uniform domains, Ann. Acad. Sci. Fenn. Ser. A I Math., 5 (1980), 197–205.
  * [30] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math., 4 (1978), 383–401.
  * [31] A. Rasila and J. Talponen On quasihyperbolic geodesics in Banach spaces. Ann. Acad. Sci. Fenn. Math., 39 (2014), 163–173.
  * [32] P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., 5 (1980), 97–114.
  * [33] J. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math., 23 (1998), 525–548.
  * [34] J. Väisälä, Quasimöbius maps, J. Analyse Math., 44 (1985), 218–234.
  * [35] J. Väisälä, Uniform domains, Tohoku Math. J., 40 (1988), 101–118.
  * [36] J. Väisälä, Free quasiconformality in Banach spaces. I, Ann. Acad. Sci. Fenn. Ser. A I Math., 15 (1990), 355–379.
  * [37] J. Väisälä, Free quasiconformality in Banach spaces. II, Ann. Acad. Sci. Fenn. Ser. A I Math., 16 (1991), 255–310.
  * [38] J. Väisälä, Free quasiconformality in Banach spaces. III, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), 393–408.
  * [39] J. Väisälä, Free quasiconformality in Banach spaces. IV, Analysis and Topology, 697-717, World Sci. Publ., River Edge, N. J., 1998\.
  * [40] J. Väisälä, Relatively and inner uniform domains, Conformal Geom. Dyn., 2 (1998), 56–88.
  * [41] J. Väisälä, The free quasiworld: freely quasiconformal and related maps in Quasiconformal geormetry and dynamics $($Lublin 1996$)$, Banach Center Publications, Vol. 48, Polish Academy of Science, Warsaw, 1999, 55-118.
  * [42] J. Väisälä, Gromov hyperbolic spaces, Expo. Math., 23 (2005), 187-231.
  * [43] X. Xie, Quasimöbius maps preserve uniform domains, Ann. Acad. Sci. Fenn. Ser. A I Math., 32 (2007), 481–495.