2 Notations and definitions

Let $f:D\subset X\rightarrow Y$ . We say that f is open if f carries open sets into open sets; we say that f is discrete if either $f^{-1}(y)=\phi $ , or $f^{-1}(y)$ is a discrete set in D, and we say that f is light if dim $f^{-1}(y)\leq 0$ for every $y\in Y$ .

If $\gamma :[0,1)\rightarrow X$ is an open path, we say that $x\in {\overline {X}}$ is a limit point of this open path if there exist $t_m\rightarrow 1$ such that $\gamma (t_m)\rightarrow x$ .

If $\alpha \geq 1$ , $E\subset D\subset X$ , and $\omega \in L_{loc}^1(D)$ , we say that $M_\omega ^\alpha (E)=0$ if $M_\omega ^\alpha (\Gamma )=0$ , where $\Gamma = \{\gamma :[0,1)\rightarrow D$ path $|\gamma $ has at least a limit point in $E\}$ . If $E=\{x\}$ , we write $M_\omega ^\alpha (x)=0$ . We present some conditions when $M_\omega ^\alpha (x)=0$ .

Let $D\subset X$ be open, let $f:D\rightarrow Y$ be continuous, open, and light, and let $p:[0,1]\rightarrow Y$ be a path and $x\in D$ such that $f(x)=p(0)$ . We say that f lifts the path p from x if there exists a path $q:[0,1]\rightarrow D$ such that $q(0)=x$ and $p=f\circ q$ . We say that $q:[0,a)\rightarrow D$ , $0<a<1$ , is a maximal lifting of p from x if $q(0)=x$ , $f\circ q=p|[0,a)$ , and q is maximal with this property. If $f:D\rightarrow Y$ is continuous, open, and light, $p:[0,1]\rightarrow Y$ is a path, $x\in D$ is such that $f(x)=p(0)$ , then either we can lift p from x, or we can find a maximal lifting $q:[0,a)\rightarrow D$ of p from x (see [Reference Stefanchuk31]).

Let X be Ahlfors upper n-regular. We say that D is a QED domain if there exists a constant $A>0$ such that $M_n(\Delta (E,F,D))\geq AM_n(\Delta (E,F,X))$ for every continua $E,F\subset D$ .

Let $h:{\overline {\mathbb {R}}}^n\rightarrow [0,\infty ]$ be the chordal metric in ${\overline {\mathbb {R}}}^n$ given by

$h(x,y)=\frac {|x-y|}{\sqrt {1+|x|^2}\sqrt {1+|y|^2}}$ if $x,y\in {\mathbb {R}}^n$ , $h(x,\infty )=\frac {1}{\sqrt {1+|x|^2}}$ if $x\in {\mathbb {R}}^n$ .

Here, $|x|=\left(\sum \limits _{i=1}^nx_i^2\right)^{\frac {1}{2}}$ if $x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n$ and if $A\subset {\overline {{\mathbb {R}}^n}}$ , we denote by $h(A)$ the diameter of A is the chordal metric. We also denote by $\mu _n$ the Lebesgue measure in ${\mathbb {R}}^n$ .

We say that a domain $A\subset {\overline {{\mathbb {R}}^n}}$ is a ring if ${\overline {{\mathbb {R}}^n}}\setminus A$ has exactly two components $C_0$ and $C_1$ , and we write $A=R(C_0,C_1)$ and we denote by $\Gamma _A=\Delta (C_0,C_1,A)$ . We know from Theorem 7.4 in [Reference Mateljevic, Salimov and Sevost’yanov21] that if $A=R(C_0,C_1)$ is a ring, there exists a constant $C(n)$ depending only on n such that

(4) $$ \begin{align}M_n(\Gamma_A)\geq\frac{\omega_{n-1}}{\left(\ln\left(\frac{C(n)}{h(C_0)h(C_1)}\right)\right)^{n-1}}.\end{align} $$

Here, $C(n)=2\lambda _n^2$ and $4\leq \lambda _n\leq 2e^{n-1}$ if $n\geq 3$ , $\lambda _2=4$ , and $(\lambda _n)^{\frac {1}{n}}\rightarrow \infty $ . If $p\geq 1$ , we set $W_{\mathrm{loc}}^{1,p}(D,{\mathbb {R}}^n)$ the Sobolev space of all mappings $f:D\rightarrow {\mathbb {R}}^n$ which are locally in $L^p$ together with their first-order distributional derivatives. If $\gamma [0,1]\rightarrow X$ is a path, we set $|\gamma |=\{x\in X|$ there exists $t\in [0,1]$ such that $x=\gamma (t)\}$ .

Let $D\subset {\mathbb {R}}^n$ be open, $n\geq 2$ , and $f:D\rightarrow {\mathbb {R}}^n$ . We say that f is of finite dilatation if $f\in W_{loc}^{1,n}(D, {\mathbb {R}}^n)$ , $J_f\in L_{loc}^1(D)$ , and there exists $K:D\rightarrow [0,\infty ] \mu _n$ -measurable and finite $\mu _n$ -a.e. such that $|f^\prime (x)|^n\leq K(x)J_f(x)$ a.e. Such a mapping has a.e. first partial derivatives, and we set $f^\prime (x)$ the matrix of the first-order partial derivatives of f at x. If $K(x)\leq K$ a.e., then f is quasiregular. We define

$$ \begin{align*}K_0(x,f)=\left\{ \begin{array}{@{}ll} \frac{|f^\prime(x)|^n}{J_f(x)},\text{ if }J_f(x)\neq 0; \\ 1,\text{ if }J_f(x)=0 \end{array} \right.\end{align*} $$

the outer dilatation of f at x. For such mappings, we define

$$ \begin{align*}K_I(x,f)=\left\{ \begin{array}{@{}ll} \frac{|D^\sharp f(x)|^n}{J_f(x)^{n-1}},\text{ if }J_f(x)> 0; \\ 1,\text{ otherwise} \end{array} \right.\end{align*} $$

the inner distortion of f at x. Here, $D^\sharp f(x)$ represents the adjoint of $f^\prime (x)$ .

We see from [Reference Hencl and Koskela16] that if $D\subset {\mathbb {R}}^n$ is a domain, $f:D\rightarrow {\mathbb {R}}^n$ is nonconstant and of finite dilatation and $K_0(\cdot ,f)\in L_{loc}^p(D)$ , where $p>n-1$ if $n\geq 3$ , $p=1$ if $n=2$ , then f is open and discrete.

3 Proofs of the results

Let $U\in {\mathcal {V}}(x)$ , $V\in {\mathcal {V}}(x)$ be such that $V\subset U\cap B(x,q)$ , $V\cap D$ is connected and $(V\cap D) \backslash E$ has n components and let $(Q_i)_{i\in {\mathbb {N}}}$ be the components of $G\setminus C(f,\partial D\cap {\overline {B}}(x,q))$ . Let $U_1,\ldots ,U_n$ be the components of $(V\cap D) \backslash $ and we first suppose that $x\in {\overline {U}}_l$ for $l=1,\ldots ,n$ . Suppose that $f(U_k)\cap Q_i\neq \phi $ and that $f(U_k)\cap \complement Q_i\neq \phi $ . Since $f(U_k)$ is connected, we find that $f(U_k)\cap \partial Q_i\neq \phi $ and we see that $\partial Q_i\subset \partial G\cup C(f, {\overline {B}}(x,q)\cap \partial D)$ . Let $z\in U_k$ be such that $f(z)\in \partial Q_i$ . Then, either $f(z)\in \partial G$ or $f(z)\in C(f, {\overline {B}}(x,q)\cap \partial D)$ . The first case cannot hold, since $f(z)\in G$ , and the second case is impossible, since $z\not \in E$ . It results that $f(U_k)\subset Q_i$ .

We proved that for every $k\in \{1,\ldots ,n\}$ , we find $i_k\in {\mathbb {N}}$ such that $f(U_k)\subset Q_{i_k}$ , the component $Q_{i_k}$ is unique, and the components $Q_{i_k}$ may not be distinct.

Let $k\in \{1,\ldots ,n\}$ be fixed and let $x_m\rightarrow x$ , $y_m\rightarrow x$ , $x_m,y_m\in U_k$ , $m\in {\mathbb {N}}$ be such that $f(x_m)\rightarrow w_1$ , $f(y_m)\rightarrow w_2$ with $w_1\neq w_2$ . Then, $w_1,w_2\in \partial Q_{i_k}$ and let $\gamma _m:[0,1]\rightarrow U_k$ be a path such that $\gamma _m(0)=x_m$ and $\gamma _m(1)=y_m$ . Since $Q_{i_k}$ is strongly accessible with respect to p-modulus at the point $w_1$ , we can find $\delta>0$ , $K\subset Q_{i_k}$ compact and $W_1\in {\mathcal {V}}(w_1)$ , $W_2\in {\mathcal {V}}(w_1)$ such that $W_2\subset W_1$ , $|f\circ \gamma _m|\cap \partial W_1\neq \phi $ , $|f\circ \gamma _m|\cap \partial W_2\neq \phi $ , and $M_p(\Delta (K,|f\circ \gamma _m|,Q_{i_k})>\delta $ for every $m\in {\mathbb {N}}$ . Let $\Gamma ^\prime _m=\Delta (K,|f\circ \gamma _m|,Q_{i_k})$ for every $m\in {\mathbb {N}}.$

We see that $f^{-1}(K)$ is closed and let $\rho =d(x,f^{-1}(K))$ . If $\rho =0$ , then there exist $\xi _l\in f^{-1}(K)$ , $l\in {\mathbb {N}}$ such that $\xi _l\rightarrow x$ and since K is compact, we can suppose that $f(\xi _l)\rightarrow z\in K$ , and hence $z\in C(f,x)\subset C(f,{\overline {B}}(x,q) \cap \partial D)$ . Then, $z\in Q_{i_k}\cap C(f,{\overline {B}}(x,q)\cap \partial D)$ and we reached a contradiction. We showed that $\rho>0$ . Let $t=d(x,\partial V)$ and $\eta =\min \{t,\rho \}$ .

Let $\Gamma _m$ be the family of all maximal liftings of some paths from $\Gamma ^\prime _m$ starting from some point of $|\gamma _m|$ for every $m\in {\mathbb {N}}$ . Let $\beta :[0,1]\rightarrow Y$ , $\beta \in \Gamma ^\prime _m$ be such that $\beta (0)\in |f\circ \gamma _m|$ and let $\alpha :[0,a)\rightarrow D$ be its maximal lifting from some point $z\in |\gamma _m|$ . Since $U_k\cap B(x,\eta )\cap f^{-1}(K)=\phi $ , we cannot lift $\alpha $ from z such that $|\alpha |\subset U_k\cap B(x,\eta )$ and it results that $|\alpha |\cap \partial (U_k\cap B(x,\eta ))\neq \phi $ . Let $0<b<a$ be such that $\alpha (s)\in U_k\cap B(x,\eta )$ for $0<s<b$ and b is maximal with this property. Let w be a limit point of $\alpha |[0,b):[0,b)\rightarrow U_k\cap B(x,\eta )$ . Since f is open, we see that $w\in \partial (U_k\cap B(x,\eta ))\subset (\partial D\cap {\overline {B}}(x,q))\cup E\cup S(x,\eta )$ . If $w\in E$ , then $f(w)\in C(f,{\overline {B}}(x,q)\cap \partial D)$ and on the other side $f(w)=\beta (b)\in Q_{i)k}$ and hence we reached a contradiction. If $w\in {\overline {B}} (x,q)\cap \partial D$ and $\partial D\cap {\overline {B}}(x,q)$ is compact, we find a point $v\in {\overline {B}}(x,q)\cap \partial D$ and $t_l\rightarrow b$ such that $\alpha (t_l)\rightarrow v$ and then $\beta (t_l)\rightarrow \beta (b)\in Q_{i_k}$ . We see that $\beta (b)\in C(f,v)\subset C(f,{\overline {B}} (x,q) \cap \partial D)$ and since $Q_{i_k}\cap C(f,{\overline {B}}(x,q)\cap \partial D)=\phi $ , we reached again a contradiction. It results that $w\in S(x,\eta )$ and since $f^{-1}(K)\cap (U_k\cap B(x,\eta ))=\phi $ , we find that $|\alpha |\cap S(x,\eta )=\phi $ . If f lifts $\beta $ from z, then $|\alpha |\cap E=\phi $ , $|\alpha |\cap C(f,{\overline {B}}(x,q)\cap \partial D)=\phi $ , and $|\alpha |\cap \partial (U_k\cap B(x,\eta ))\neq \phi $ and we also find that $|\alpha |\cap S(x,\eta )\neq \phi $ .

We can suppose that $|\gamma _m|\subset B(x,r_m)$ with $r_m\rightarrow 0$ , $r_m<\eta $ for $m\in {\mathbb {N}}$ , and then $\Gamma _m>\Gamma _{x,r_m,\eta ,D}$ for every $m\in {\mathbb {N}}$ and we also have that $\Gamma ^\prime _m>f(\Gamma _m)$ for $m\in {\mathbb {N}}$ . Since $M_\omega ^\alpha (x)=0$ , we see from Lemma 2.5 in [Reference Cristea5] that $M_\omega ^\alpha (\Gamma _{x,r_m,\eta ,D})\rightarrow 0$ when $m\rightarrow \infty $ . We have $\delta \leq M_p(\Gamma ^\prime _m)\leq M_p(f(\Gamma _m))\leq M_p(f(\Gamma _{x,r_m,\eta ,D}))\leq M_\omega ^\alpha (\Gamma _{x,r_m,\eta ,D}))\rightarrow 0$ if $m\rightarrow \infty $ .

We reached a contradiction.

We proved that there exists $\lim _{\substack {y\rightarrow x\\ y\in U_k}}f(y)=z_k\in {\overline {Y}}$ for $k=1,\ldots ,n$ .

Since $V\cap D$ is connected for arbitrary small $V\in {\mathcal {V}}(x)$ , we see that $C(f,x)$ is connected. Let $\xi _m\in V$ , $\xi _m\rightarrow x$ . Since $\mathrm{Int} \, E=\phi $ , we can find $\eta _m\in (V\cap D) \setminus E$ such that $d(\xi _m,\eta _m)<\frac {1}{m}$ and $d(f(\xi _m),f(\eta _m))<\frac {1}{m}$ for every $m\in {\mathbb {N}}$ . Let $I_k=\{m\in {\mathbb {N}}|\eta _m\in U_k\}$ for $k=1,\ldots ,n$ and let $1\leq l\leq n$ be such that $I_l$ is infinite. Then, $f(\eta _m)\rightarrow z_l$ if $m\rightarrow \infty $ , $m\in I_l$ and hence $f(\xi _m)\rightarrow z_l$ if $m\rightarrow \infty $ , $m\in I_l$ . It results that $C(f,x)\subset \{z_1,\ldots ,z_n\}$ and since $C(f,x)$ is connected, we see that $C(f,x)=\{y\}$ and $\lim _{z\rightarrow x}f(z)=\{y\}$ .

In the case when $x\not \in U_l$ for $l=1,\ldots ,n$ , we take $W\in {\mathcal {V}}(x)$ , $W\subset V$ such that $W\cap E=\phi $ and $W\cap D$ is connected. Then, $f(W\cap D)$ is connected and there exists a single component $Q_i$ such that $f(W\cap D)\subset Q_i$ . Using the same arguments, we show that there exists $\lim _{y\rightarrow x}f(y)\in {\overline {Y}}$ .

Let $\epsilon>0$ . There exists $\delta _\epsilon>0$ such that $f(B(x_0,\delta _\epsilon )\cap D)\subset B(C(f,x),\epsilon )$ and taking $\epsilon $ small enough, we find for every $k\in \{1,\ldots ,m(r)\}$ a continuum $K_k$ such that $h(K_k)>\delta _k>0$ , $f(B(x_0,\delta _\epsilon )\cap D)\cap K_k=\phi $ , $K_k\subset Q_{i_k}$ and as before, we see that $\rho _k=d(x_0,f^{-1}(K_k))>0$ .

Let $\lambda <r$ . Then, $B(x_0,\lambda )\cap D) \backslash E$ has $V_1,\ldots ,V_{m(\lambda )}$ components and for every $V_i$ there exists a component $Q_{i_k}$ such that $f(V_i)\subset Q_{i_k}$ , $i=1,\ldots ,m(\lambda )$ and $k\in \{1,\ldots ,m(r)\}$ . We take $\lambda <\rho _k$ , $k=1,\ldots ,m(r)$ and then $f^{-1}(K_k)\cap B(x_0,\lambda )=\phi $ for every $k=1,\ldots ,m(r)$ .

It results that taking $r>0$ small enough such that $D\cap B(x_0,r)$ is connected, $(D\cap B(x_0,r)) \backslash E$ has $U_1,\ldots ,U_m$ components, we find $Q_{i_1},\ldots ,Q_{i_k}$ components (which may be not distinct) such that $f(U_k)\subset Q_{i_k}$ , a continuum $K_k$ such that $h(K_k)> \delta _k>0$ , $K_k\subset Q_{i_k}$ , $f(B(x,r)\cap D)\cap K_k=\phi $ and $r<\rho _k$ for $k=1,\ldots ,m$ .

Let $k\in \{1,\ldots ,m\}$ be fixed and $y,z\in B(x_0,r)\cap D$ and let $\gamma _k:[0,1] \rightarrow B(x_0,r)\cap D$ be a path such that $\gamma _k(0)=y$ and $\gamma _k(1)=z$ . Let $\Gamma ^\prime _k=\Delta (|f\circ \gamma _k|,K_k,Q_{i_k})$ . Since $Q_{i_k}$ is a QED domain, there exists $A_k>0$ such that $M_n(\Gamma _k^\prime )\geq A_kM_n(|f\circ \gamma _k|,K_k,{\overline {{\mathbb {R}}^n}})$ .

Let $\Gamma _k$ be the family of all maximal liftings of some paths from $\Gamma ^\prime _k$ starting from some point of $|\gamma _k|$ . Since $f^{-1}(K_k)\cap B(x_0,r)=\phi $ , we see as before that if $\beta \in \Gamma _k$ , then $|\beta |\cap \partial B(x_0,r)\neq \phi $ . Let us take the points $y,z$ and the path $\gamma _k$ such that $|\gamma _k|\subset D\cap B(x_0,r_k)$ with $0<r_k<r$ . Then, $\Gamma>\Gamma _{x_0,r_k,r,D}$ and $\Gamma ^\prime>f(\Gamma )$ and using Theorem D and relation (4), we have

$$ \begin{gather*}\frac{A_k\omega_{n-1}}{\left(\ln\left(\frac{C(n)}{\delta_kh(f(y),f(z))}\right)\right)^{n-1}}\leq\frac{A_k\omega_{n-1}}{\left(\ln\left(\frac{C(n)}{h(|f\circ\gamma_k|)h(K_k)}\right)\right)^{n-1}}\leq\\\leq A_kM_n(\Delta(|f\circ \gamma_k|,K_k,{\overline{{\mathbb{R}}^n}}))\leq M_n(\Gamma_k^\prime)\leq M_n(f(\Gamma_k))\leq M_\omega^n(\Gamma_{x_0,r_k,r,D}) \leq\frac{C\omega_{x_0}}{(\ln\ln(\frac{er}{r_k}))^n}.\end{gather*} $$

It results that if $y,z\in U_k$ , then

(5) $$ \begin{align}h(f(y),f(z))\leq\frac{C(n)}{\delta_k}\exp\left(-\mu_k\left(\ln\ln\left(\frac{er}{r_k}\right)\right)^{\frac{n}{n-1}}\right)\leq\frac{C(n)}{\delta_k}\left(\frac{r_k}{er}\right)^{\mu_k}. \end{align} $$

Here, $\mu _k=(\frac {A_k\omega _{n-1}}{C\omega _{x_0}})^{\frac {1}{n-1}}$ .

Letting $y_p,z_p\rightarrow x_0$ and $(r_k)_p\rightarrow 0$ in relation (5), we find that there exists $\lim _{\substack {x\rightarrow x_0\\ x\in U_k}} f(x)\in {\overline {{\mathbb {R}}^n}}$ and we also denote this limit $f(x_0)$ .

Letting $y\rightarrow x$ , $z\rightarrow x_0$ with $x\in S(x_0,r_k)$ in relation (5), we obtain

(6) $$ \begin{align}h(f(x),f(x_0))\leq\frac{C(n)}{\delta_k}\left(\frac{d(x,x_0)}{er}\right)^{\mu_k}\text{ if }d(x,x_0)<r\text{ and }x\in{\overline{U}}_k.\end{align} $$

Taking $\delta =\min \{\delta _1,\ldots ,\delta _m\}$ , $A=\min \{A_1,\ldots ,A_m\}$ , and $\mu =(\frac {A\omega _{n-1}}{C\omega _{x_0}})^{\frac {1}{n-1}}$ , we find that

(7) $$ \begin{align}h(f(x),f(x_0))\leq\frac{C(n)}{\delta}\left(\frac{d(x,x_0)}{er}\right)^\mu\text{ if } x\in D\cap B(x_0,r).\end{align} $$

Remark A Suppose that we have in addition the following condition in Theorem 2, in connection with Theorem 6.1 in [Reference Desyatka, Kovba and Sevost’yanov10]:

a) There exists $r_0>0$ and constant $C_1$ such that for every $0<r<r_0$ and every $y,z\in D\cap B(x_0,r)$ , there exists a path $\gamma :[0,1]\rightarrow D\cap B(x_0,r)$ such that $\gamma (0)=y$ , $\gamma (1)=z$ and $d(\gamma (t),x_0)\leq C_1\max \{d(y,x_0),d(z,x_0)\}$ for every $t\in [0,1]$ . Then, there exists a constant $C_2$ and $r>0$ small enough such that

(8) $$ \begin{align}h(f(y),f(z))\leq C_2\max\{d(y,x_0)^\mu,d(z,x_0)^\mu\}\text{ if }y,z\in D\cap B(x_0,r). \end{align} $$

Proof We see from relations (5) and (6) that, keeping the notations from Theorem 2, we have

(9) $$ \begin{align}h(f(y),f(x))\leq\frac{C(n)}{\delta}\left(\frac{\max\{d(x,x_0)^\mu,d(y,x_0)^\mu\}}{er}\right)\text{ if }y,x\in{\overline{U}}_k,k=1,\ldots,m \end{align} $$

and $r>0$ is small enough.

We can take $0<\mu <1$ .

Let $r>0$ be such that relation (8) holds for $k=1,\ldots ,m$ and let $y,z\in D\cap B(x_0,r)$ . Let $\gamma :[0,1]\rightarrow D\cap B(x_0,r)$ be a path such that $\gamma (0)=y$ , $\gamma (1)=z$ , and $d(\gamma (t),x_0)\leq C_1\max \{d(y,x_0),d(z,y_0)\}$ . Suppose that $t_1=0$ , $y\in U_1$ and let $t_2=\inf \{t>0|\gamma (t)\in \partial U_1\}$ . If $t_2=1$ , then the process ends. If $t_2<1$ , let $t_3=\sup \{t>t_2|\gamma (t)\in \partial U_1\}$ . If $t_3=1$ , then the process ends. If $t_3<1$ , then we can suppose that $\gamma (t)\in U_2$ for some $t>t_3$ and let $t_4=\inf \{t>t_3| \gamma (t)\in \partial U_2\}$ . If $t_4=1$ , then the process ends. If $t_4<1$ , let $t_5=\sup \{t>t_4|\gamma (t)\in \partial U_2\}$ . We continue the process which ends in $2p$ steps, $p\leq n$ and $\gamma (t_{2p})=z$ . Using (9), we have $h(f(\gamma (t_{l})),f(\gamma (t_{l+1})))\leq \frac {C(n)}{\delta }\left(\frac {\max \{d(\gamma (t_{l}),x_0),d(\gamma (t_{l+1}),x_0)\}}{er}\right)^\mu $ , $l=1,\ldots ,2p-1$ .

Then,

$$ \begin{gather*}\hspace{-8pt}h(f(y),f(z))\leq\sum\limits_{l=1}^{2p-1}h(f(\gamma(t_{l})),f(\gamma(t_{l+1}))\leq\\\leq\frac{C(n)}{\delta}\sum\limits_{l=1}^{2p-1}\left(\frac{\max\{d(\gamma(t_{l}),x_0),d(\gamma(t_{l+1}),x_0)\}}{er}\right)^\mu\leq\\\leq\text{(using the fact that }g^{\prime\prime}(t)<0)\text{ for }t>0,\text{ where }g:{\mathbb{R}}\rightarrow {\mathbb{R}}\text{ is given by }g(t)=t^\mu\\\text{for }t>0\text{ and Jensen's inequality)}\leq\\\leq\frac{(2p)^{1-\mu}C(n)}{\delta(er)^\mu}\left(\sum\limits_{l=1}^{2p-1}\max\{d(\gamma(t_{l}),x_0),d(\gamma(t_{l+1}),x_0)\}\right)^\mu\leq\\\leq\frac{C(n)(2m)^{2-\mu}}{\delta(er)^\mu}d(\gamma(t),x_0)^\mu(\text{for some }t\in[0,1])\leq\\\leq\frac{C(n)C_1(2m)^{2-\mu}}{\delta(er)^\mu}(\max\{d(z,x_0),d(y,x_0)\})^\mu=\frac{C(n)C_1(2m)^{2-\mu}}{\delta(er)^\mu}\max\{d(z,x_0)^\mu,d(y,x_0)^\mu\}\end{gather*} $$

if $y,z\in D\cap B(x_0,r)$ .

Taking $C_2=\frac {C(n)C_1(2m)^{2-\mu }}{\delta (er)^\mu }$ , we prove relation (8).

Remark 5 If $B(x_0,r)\cap D$ is convex for every $0<r<r_0$ , then there exists $r>0$ and $0<\mu <1$ such that

$$ \begin{align*}h(f(y),f(z))\leq C_2\max\{|y-x_0|^\mu,|z-x_0|^\mu\}\text{ if }y,z\in D\cap B(x_0,r).\end{align*} $$

Also, Theorems 3 and 4 are valid for quasiregular mappings.