2 Preliminaries

This section establishes a foundation for our subsequent discussions by introducing essential notations and definitions.

Let sh, ch, th, and arth denote the hyperbolic functions $\sinh $ , $\cosh $ , $\tanh $ , and $\mathrm {arctanh}$ , respectively. Consider the Euclidean space $\mathbb {R}^n$ with $n \geq 2$ and define $\mathbb {H}^n = \{x=(x_1, \ldots , x_n) \in \mathbb {R}^n : x_n> 0\}$ as the Poincaré half-space or the upper half-plane. The ball with center x in $\mathbb {R}^n$ and radius $r>0$ is denoted as $B^n(x,r)$ , defined as the set $\{y\in \mathbb {R}^n:|y-x|<r\}$ . Correspondingly, the sphere sharing the same center and radius is $S^{n-1}(x,r)=\{y\in \mathbb {R}^n:|y-x|=r\}$ . The unit ball will be denoted by $\mathbb {B}^n=B^n(0,1)$ . Also, $\overline {B}^n(x,r)=\{y\in \mathbb {R}^n:|y-x|\leq r\}$ . For any x within a domain G in $\mathbb {R}^n$ , the Euclidean distance $d_G(x)$ is defined as the minimum distance from x to the boundary of G, denoted by $d_G(x)=d(x,\partial G)=\inf \{|x-w|: w\in \partial G\}$ . In the hyperbolic space $\mathbb {H}^n$ , the hyperbolic distance $\rho $ is characterized by the differential equation $\mathrm {d}\rho =|\mathrm {d}x|/x_n$ . Explicit formulas for the distances between points in both the upper half-space $\mathbb {H}^n$ and the unit ball $\mathbb {B}^n$ , respectively, are as follows (see [Reference Hariri, Klén and Vuorinen10, (4.8), p. 52; (4.16), p. 55]):

$$ \begin{align*} \mathrm{ch} \rho_{\mathbb{H}^n}(x,y)= 1+\frac{|x-y|^2}{2d_{\mathbb{H}^n}(x)d_{\mathbb{H}^n}(y)},\quad x,y\in \mathbb{H}^n \end{align*} $$

and

$$ \begin{align*} \mathrm{sh}^2 \frac{\rho_{\mathbb{B}^n}(x,y)}{2}=\frac{|x-y|^2}{(1-|x|^2)(1-|y|^2)},\quad x,y\in \mathbb{B}^n. \end{align*} $$

The quasihyperbolic distance, denoted as $k_G(x, y)$ , between points x and y in the domain G, is formally defined as the infimum of the integral along rectifiable curves $\gamma \subset G$ containing both x and y. This integral is calculated as the quotient of the absolute value of the differential element $\mathrm {d}x$ by the distance function $d_G(x)$ , as given by the expression:

$$ \begin{align*} k_G(x,y)=\inf_{\gamma} \int_{\gamma}\frac{|\mathrm{d}x|}{d_G(x)}. \end{align*} $$

Gehring and Palka introduced the metric $k_G(x,y)$ in [Reference Gehring and Palka7, p. 173] and provided a proof for the sharp inequalities (see [Reference Gehring and Palka7, Lemma 2.1]). These inequalities are expressed as follows:

(2.1) $$ \begin{align} k_G(x,y)\geq \left|\log \frac{d_G(x)}{d_G(y)}\right| \end{align} $$

and

(2.2) $$ \begin{align} k_G(x,y)\geq \log \left(1+\frac{|x-y|}{d_G(x)}\right). \end{align} $$

For a detailed discussion, we refer to [Reference Hariri, Klén and Vuorinen10], p. 68. It is well-known that (see [Reference Gehring and Palka7], p. 174)

(2.3) $$ \begin{align} k_{\mathbb{H}^n}(x,y)=\rho_{\mathbb{H}^n}(x,y)\quad \mathrm{and}\quad k_{\mathbb{B}^n}(x,y) \leq \rho_{\mathbb{B}^n}(x,y) \leq 2k_{\mathbb{B}^n}(x,y). \end{align} $$

For any open set $\Omega $ in $\mathbb {R}^n$ , where $\Omega $ is not equal to the entire space $\mathbb {R}^n$ , the distance ratio metric is defined by

$$ \begin{align*} j_\Omega(x, y)=\log\left(1+\frac{|x-y|}{\min\{d_\Omega(x),d_\Omega(y)\}} \right),\quad x, y\in \Omega. \end{align*} $$

When $\Omega \in \{\mathbb {B}^n, \mathbb {H}^n\}$ as per [Reference Hariri, Klén and Vuorinen10, Lemma 4.9], the following double-inequality holds:

(2.4) $$ \begin{align} j_\Omega(x,y)\leq \rho_{\Omega}(x,y)\leq 2 j_\Omega(x,y). \end{align} $$

Modulus of a curve family. Let $\Gamma $ be a family of curves in $\mathbb {R}^n$ . Also, let $\mathcal {F}(\Gamma )$ denote the family of all nonnegative Borel-measurable functions $\sigma :\mathbb {R}^n\to \mathbb {R} \cup \{\infty \}$ such that $\int _\gamma \sigma \mathrm {d}\tau \geq 1$ for each locally rectifiable curve $\gamma \in \Gamma $ . The modulus of a curve family $\Gamma \subset \mathbb {R}^n$ is defined by (see [Reference Hariri, Klén and Vuorinen10], p. 104)

$$ \begin{align*} {\mathsf{M}}(\Gamma)=\inf_{\sigma\in\mathcal{F}(\Gamma)}\int_{\mathbb{R}^n}\sigma^n \mathrm{d} m, \end{align*} $$

where m stands for the n-dimensional Lebesgue measure.

We denote by $\Delta (E,F;G)$ the family of all closed nonconstant curves joining two non-empty sets E and F in a domain G, where E, F, and G are subsets of $\overline {\mathbb {R}}^n$ .

Modulus metric. Let G be a proper subdomain of $\overline {\mathbb {R}}^n$ . The modulus metric is defined by

$$ \begin{align*} \mu_G(x,y)=\inf_{C_{xy}}{\mathsf{M}}(\Delta(C_{xy},\partial G;G)), \end{align*} $$

where the infimum is taken over all continuous paths $C_{xy}$ in G joining x and y, represented by a continuous function $\gamma :[0,1]\rightarrow G$ satisfying $\gamma (0)=x$ and $\gamma (1)=y$ . The definition of modulus metric is illustrated in Figure 1.

Figure 1

Conformal invariant $\mu _G(x,y)$ .

Uniformity. (See [Reference Hariri, Klén and Vuorinen10, Definition 6.1]) A domain G of $\mathbb {R}^n$ , where $G\neq \mathbb {R}^n$ , is termed uniform if there exists a constant $A=A(G)\geq 1$ such that $k_G(x,y)\leq A j_G(x,y)$ for all $x,y\in G$ . The unit ball $\mathbb {B}^n$ and the upper half-space $\mathbb {H}^n$ are examples of uniform domains with the constant $2$ , as implied by (2.3) and (2.4), respectively.

Absolutely continuous on lines (ACL). Consider $\mathbb {R}^{n-1}_j$ as the set $\mathbb {R}^{n-1}_j=\{x\in \mathbb {R}^n: x_j=0\}$ , where $j=1,2,\ldots ,n$ . Suppose that $T_j:\mathbb {R}^n\rightarrow \mathbb {R}^{n-1}_j$ is an onto orthogonal projection $T_j x=x-x_je_j$ and $Q=\{x\in \mathbb {R}^n: a_j\leq x_j\leq b_j\}$ is a closed n-interval. A mapping $\phi : Q \rightarrow \mathbb {R}$ is called absolutely continuous on lines, abbreviated as ACL, if it is absolutely continuous on almost every line segment in Q, parallel to the coordinate axes $e_1,\ldots ,e_n$ . More precisely, if $E_j$ is the set of all $x\in T_j Q$ such that the mapping $t\mapsto \phi (x+te_j)$ is not absolutely continuous on $[a_j,b_j]$ , then $m_{n-1}(E_j)=0$ for all $j=1,\ldots ,n$ .

For an open set $\Omega $ in $\mathbb {R}^n$ , an ACL mapping $\phi :\Omega \rightarrow \mathbb {R}$ is said to be ACL $^n$ , $n\geq 1$ , if $\phi $ is locally $L^n$ -integrable in $\Omega $ and if the partial derivatives $\partial _j \phi $ (which exist a.e. and are measurable) of $\phi $ are locally $L^n$ -integrable as well (see [Reference Martio, Ryazanov, Srebro and Yakubov20, p. 22]).

Quasiregular mappings. Consider a domain $G\subset \mathbb {R}^n$ . A mapping $f: G \rightarrow \mathbb {R}^n$ is said to be K-quasiregular if f belongs to ACL $^n$ and if there exists a constant $K\geq 1$ satisfying the inequality

$$ \begin{align*} |f'(x)|^n\leq K J_f(x),\quad \mathrm{where}\quad|f'(x)|=\max_{|\phi|=1}|f'(x)\phi|, \end{align*} $$

almost everywhere in G. Here, $f'(x)$ and $J_f(x)$ represent the formal derivative and the Jacobian determinant of f at the point x, respectively.

Quasiconformal mappings. Let G, $G'$ be domains in $\overline {\mathbb {R}}^n=\mathbb {R}^n\cup \{\infty \}$ , $K\geq 1$ and let $f:G\rightarrow G'$ be a homeomorphism. Then, f is K-quasiconformal if and only if the following conditions are satisfied:

• • f is ACL $^n$ ;

• • f is differentiable;

• • for almost all $x\in G$

  $$ \begin{align*} |f'(x)|^n/K\leq |J_f(x)|\leq K L (f'(x))^n, \end{align*} $$
  where $L(\lambda )=\min _{|\phi |=1}|\lambda \phi |$ .

The Harnack inequality provides a basis for defining a Harnack (pseudo) metric. Consider $\mathcal {H}^+(G)$ as the class of all positive harmonic functions u in G.

Harnack metric. For arbitrary $x, y \in G$ , the Harnack metric is defined by

$$ \begin{align*} h_G(x,y)=\sup\left|\log \frac{u(x)}{u(y)}\right|, \end{align*} $$

where the supremum is taken over all $u\in \mathcal {H}^+(G)$ . This metric has been investigated in various contexts, including studies in [Reference Bear and Smith3, Reference Chirka5, Reference Herron and Minda15, Reference Herron14, Reference Köhn19, Reference Shiga22].

3 $(s, C(s))$ -Harnack functions and Harnack metric

In this section, we present our results on $(s, C(s))$ -Harnack functions and the Harnack metric under K-quasiconformal and K-quasiregular mappings. We start with the following:

Lemma 3.1 All positive harmonic functions on $B^n(x,r)\subset \mathbb {R}^n$ are $(s, C(s))$ -Harnack with

$$ \begin{align*} C(s)=C(s,n)=\frac{1}{1-s^2}\left(\frac{1+s}{1-s}\right)^n \end{align*} $$

for all $s\in (0,1)$ .

Proof Let u be any positive harmonic function on $B^n(x,r)$ and $0<\delta <r$ . Then, by [Reference Helms13, Theorem 3.2.1], we have

(3.1) $$ \begin{align} \frac{u(x_1)}{u(x_2)}\leq \frac{r^2}{r^2-\delta^2}\left(\frac{r+\delta}{r-\delta}\right)^n \end{align} $$

for all $x_1,x_2\in B^n(x,\delta )$ . Now, it is enough to put $\delta =r s$ in (3.1) since $rs<r$ for all $s\in (0,1)$ .

Theorem 3.2 (i) Let $s\in (0,1)$ and $u:\mathbb {B}^{n} \rightarrow (0,\infty )$ be a Harnack function. Then for all $x,y\in \mathbb {B}^n$

$$ \begin{align*} u(x)\leq C(s)^{1+t} u(y), \qquad t=\frac{\log((1+r)/(1-r))}{\log((1+s)/(1-s))}, \end{align*} $$

where $r=\mathrm {th}(\rho _{\mathbb {B}^n}(x,y)/2)$ and $C(s)\geq 1$ . (ii) If u is a positive harmonic function, $x\in \mathbb {B}^n$ , $s\in (0,1)$ and $y\in S^{n-1}(x, s(1-|x|))$ , then

$$ \begin{align*} u(x)\leq \frac{1}{1-s^2}\left(\frac{1+s}{1-s}\right)^n u(y), \qquad s<\exp(\rho_{\mathbb{B}^n}(x,y))-1. \end{align*} $$

Proof (i) The proof follows from Definition 1.2 and [Reference Hariri, Klén and Vuorinen10, Lemma 6.23]. (ii) It follows from [Reference Hariri, Klén and Vuorinen10, Lemma 4.9(1)] that

$$ \begin{align*} \rho_{\mathbb{B}^n}(x,y)\geq j_{\mathbb{B}^n}(x,y)\geq \log\left(1+\frac{s(1-|x|)}{1-|x|}\right)=\log(1+s). \end{align*} $$

This completes the proof.

We continue with the following result on quasiregular mappings; in fact, we show that if $f: G\rightarrow \mathbb {R}^n$ is a quasiregular mapping, and if $\partial fG$ satisfies some additional conditions, then the function $u(x)=d_{fG}(f(x))$ , $(x\in G)$ , satisfies the $(s, C(s))$ -Harnack inequality.

Theorem 3.4 Let G be a proper subdomain of $\mathbb {R}^n$ , and let $f: G\rightarrow \mathbb {R}^n$ be a K-quasiregular mapping such that $fG\subset \mathbb {R}^n$ is a A-uniform domain. Also, let $\partial fG$ be connected such that it consists of at least two points. Then, the function $u(x)=d_{fG}(f(x))$ , $(x\in G)$ , satisfies the $(s, C(s))$ -Harnack inequality with the constant

(3.2) $$ \begin{align} C(s)=\exp\left(\frac{A K_I(f)}{c_n}\omega_{n-1}\left(\log \frac{s d_G(x)}{|x-y|}\right)^{1-n}\right),\quad s\in(0,1), \end{align} $$

for $y\in B_{x,s}={B}^n(x,s d_G(x))$ , where $\omega _{n-1}$ is the $(n-1)$ -dimensional surface area of $S^{n-1}$ , $K_I(f)$ is the inner dilatation of f, and $c_n$ is a constant number depending only on n.

Proof Since $\partial fG$ is a connected domain and $fG$ is a A-uniform domain, by [Reference Hariri, Klén and Vuorinen10, Lemma 10.8(1)] and by definition, we have

(3.3) $$ \begin{align} \mu_{fG}(f(x),f(y))\geq c_n j_{fG}(f(x),f(y))\geq \frac{c_n}{A}k_{fG}(f(x),f(y)),\quad x,y\in G, \end{align} $$

where $A\geq 1$ , and $c_n$ is a constant number depending on n. Also, by [Reference Hariri, Klén and Vuorinen10, Theorem 15.36(1)], the following inequality

(3.4) $$ \begin{align} \mu_{fG}(f(x),f(y))\leq K_I(f)\mu_G(x,y),\quad x,y\in G \end{align} $$

holds for a nonconstant quasiregular mapping $f: G\rightarrow \mathbb {R}^n$ , where $K_I(f)\geq 1$ is the inner dilatation of f. It follows from [Reference Hariri, Klén and Vuorinen10, Lemma 10.6(2)] that if $x\in G$ and $y\in B_{x,s}={B}^n(x,s d_G(x))$ with $x\neq y$ , then

(3.5) $$ \begin{align} \mu_G(x,y)\leq \mu_{B_{x,s}}(x,y)\leq \omega_{n-1}\left(\log \frac{1}{r} \right)^{1-n}, \end{align} $$

where $r=|x-y|/(sd_G(x))$ . Now, by (2.1) and (3.3)–(3.5), we obtain

$$ \begin{align*} \left|\log \frac{d_{fG}(f(x))}{d_{fG}(f(y))}\right| &\leq k_{fG}(f(x),f(y))\leq \frac{A}{c_n}\mu_{fG}(f(x),f(y))\leq \frac{A K_I(f)}{c_n}\mu_{G}(x,y) \\ &\leq \frac{AK_I(f)}{c_n}\mu_{B_{x,s}}(x,y)\leq \frac{AK_I(f)}{c_n}\omega_{n-1}\left(\log \frac{s d_G(x)}{|x-y|}\right)^{1-n}. \end{align*} $$

This establishes the desired inequality (3.2), and thus concludes the proof.

Remark 3.5 In Theorem 3.4, the connectedness of $\partial fG$ is crucial. However, it is noteworthy that the statement of Theorem 3.4 can be invalidated by the existence of an analytic function $f:\mathbb {B}^2\rightarrow \mathbb {B}^2\,{\backslash}\, \{0\}=f\mathbb {B}^2$ . An explicit example of such a function is defined by $f:\mathbb {B}^2\rightarrow \mathbb {B}^2\,{\backslash}\, \{0\}$ as

$$ \begin{align*} f(z)=\exp\left(\frac{z+1}{z-1}\right),\quad z\in \mathbb{B}^2. \end{align*} $$

Let $x_p = (e^p - 1)/(e^p + 1)$ for $p = 1, 2, \ldots $ . Considering $f(x_p) = \exp (-e^p)$ and $f(x_{p+1}) = \exp (-e^{p+1})$ , we can deduce

$$ \begin{align*} \left|\frac{f(x_p)}{f(x_{p+1})}\right| = \frac{\exp(e^{p+1})}{\exp(e^p)}. \end{align*} $$

Additionally, employing a straightforward calculation, we can infer from (2.2) that

$$ \begin{align*} \left(\log \frac{sd(x_p)}{|x_p - x_{p+1}|}\right)^{1-n} \leq \left(\log \frac{s}{\exp(k_{\mathbb{B}^2}(x_p,x_{p+1}))-1}\right)^{1-n}. \end{align*} $$

Moreover, due to $k_{\mathbb {B}^2}(x,y)\leq 2 j_{\mathbb {B}^2}(x,y)$ , the preceding inequality leads to

$$ \begin{align*} \left(\log \frac{sd(x_p)}{|x_p-x_{p+1}|}\right)^{1-n}\leq \left(\log \frac{s}{\exp(2j_{\mathbb{B}^2}(x_p,x_{p+1}))-1}\right)^{1-n}. \end{align*} $$

Finally, by applying Theorem 3.4 and utilizing (2.4), we derive

$$ \begin{align*} \frac{\exp(e^{p+1})}{\exp(e^p)}\leq \exp\left(\frac{A K_I(f)}{c_n}\omega_{n-1}\left(\log \frac{s}{\exp(2\rho_{\mathbb{B}^2}(x_p,x_{p+1}))-1}\right)^{1-n}\right). \end{align*} $$

As $\rho _{\mathbb {B}^2}(x_p,x_{p+1})=1$ , the right-hand side of the last inequality remains bounded. However, the left-hand side of the same inequality diverges to infinity as p approaches infinity. Consequently, we can infer that the assertion in Theorem 3.4 loses validity when $\partial fG$ includes isolated points.

In the following, we shall study the Harnack metric $h_G(x,y)$ , where G is a proper subdomain of $\mathbb {R}^n$ .

Theorem 3.6 Let $s\in (0,1)$ and $C(s)\geq 1$ . (i) If G is a proper subdomain of $\mathbb {R}^n$ , then

$$ \begin{align*} h_G(x,y)\leq \left(1+\frac{k_G(x,y)}{2\log(1+s)}\right)\log C(s). \end{align*} $$

(ii) If $G=\mathbb {B}^n$ or $G=\mathbb {H}^n$ , then we have

$$ \begin{align*} h_G(x,y)\leq \left(1+\frac{\rho_G(x,y)}{\log[(1+s)/(1-s)]}\right)\log C(s). \end{align*} $$

Proof (i) Let $u:G\rightarrow (0,\infty )$ be a Harnack function. By [Reference Hariri, Klén and Vuorinen10, Lemma 6.23], we have

$$ \begin{align*} \frac{u(x)}{u(y)}\leq C(s)^{1+t}\Leftrightarrow \log \frac{u(x)}{u(y)}\leq (1+t)\log C(s), \end{align*} $$

where $t=k_G(x,y)/(2\log (1+s))$ . The claim is now a direct consequence of the Harnack metric definition. (ii) According to [Reference Hariri, Klén and Vuorinen10, Lemma 6.23], the proof closely resembles that of part (i), so we skip the details.

To prove the next results, the following two lemmas will be helpful.

Lemma 3.7 [Reference Bear2, Corollary 1] For all $x,y\in \mathbb {B}^n$ ,

$$ \begin{align*} h_{\mathbb{B}^n}(x,y)=2\rho_{\mathbb{B}^n}(x,y). \end{align*} $$

Lemma 3.8 [Reference Bear and Smith3, Lemma 2.5] If $x,y\in \mathbb {H}^n$ , then

$$ \begin{align*} h_{\mathbb{H}^n}(x,y)=\rho_{\mathbb{H}^n}(x,y). \end{align*} $$

Theorem 3.9 (i) If $f: \mathbb {B}^n \rightarrow f \mathbb {B}^n$ is a nonconstant K-quasiregular mapping with $f\mathbb {B}^n \subset \mathbb {B}^n$ , then the inequality

$$ \begin{align*} h_{f\mathbb{B}^n}(f(x),f(y))\leq 2 K(h_{\mathbb{B}^n}(x,y)/2+\log4) \end{align*} $$

holds for all $x, y \in \mathbb {B}^n$ .

(ii) If $f: \mathbb {B}^n \rightarrow f \mathbb {B}^n=\mathbb {B}^n$ is a K-quasiconformal mapping, then the inequality

$$ \begin{align*} h_{f\mathbb{B}^n}(f(x),f(y))\leq b \max\{h_{\mathbb{B}^n}(x,y), 2^{1-\alpha}h_{\mathbb{B}^n}(x,y)^\alpha\} \end{align*} $$

holds, where $\alpha =K^{1/(1-n)}$ and b is a constant depending on K and n. Here, b tends to $1$ as K tends to $1$ .

Proof (i) By [Reference Hariri, Klén and Vuorinen10, Theorem 16.2(2)], we have

(3.6) $$ \begin{align} \rho_{f\mathbb{B}^n}(f(x),f(y))\leq K(\rho_{\mathbb{B}^n}(x,y)+\log 4) \end{align} $$

for all $x,y\in \mathbb {B}^n$ , where $f:\mathbb {B}^n\rightarrow f\mathbb {B}^n\subset \mathbb {B}^n$ is a K-quasiregular mapping. It follows also from Lemma 3.7 that, for $x,y\in \mathbb {B}^n$

(3.7) $$ \begin{align} h_{f\mathbb{B}^n}(f(x),f(y))= 2 \rho_{f\mathbb{B}^n}(f(x),f(y)). \end{align} $$

Now, combining (3.7) and (3.6) with Lemma 3.7 gives the desired result. (ii) Let $f: \mathbb {B}^n \rightarrow f \mathbb {B}^n=\mathbb {B}^n$ be a K-quasiconformal mapping and $x,y\in \mathbb {B}^n$ . Then, by Corollary 18.5 in [Reference Hariri, Klén and Vuorinen10], we have

(3.8) $$ \begin{align} \rho_{\mathbb{B}^n}(f(x),f(y))\leq b \max\{\rho_{\mathbb{B}^n}(x,y), \rho_{\mathbb{B}^n}(x,y)^\alpha\}, \end{align} $$

where $\alpha =K^{1/(1-n)}$ and b is a constant depending on K and n. Now, by (3.8), and using Lemma 3.7, the conclusion is obtained.

Theorem 3.10 Let $f:\mathbb {H}^n\rightarrow \mathbb {H}^n$ be a nonconstant K-quasiregular mapping such that $f\mathbb {H}^n\subset \mathbb {H}^n$ . Then

$$ \begin{align*} h_{f\mathbb{H}^n}(f(x),f(y))\leq K(h_{\mathbb{H}^n}(x,y)+\log 4), \end{align*} $$

where $K\geq 1$ .

Proof If $f:\mathbb {H}^n\rightarrow \mathbb {H}^n$ is a nonconstant K-quasiregular mapping such that $f\mathbb {H}^n\subset \mathbb {H}^n$ , then by [Reference Hariri, Klén and Vuorinen10, Theorem 16.2(2)], we have

(3.9) $$ \begin{align} \rho_{f\mathbb{H}^n}(f(x),f(y))\leq K (\rho_{\mathbb{H}^n}(x,y)+\log 4), \end{align} $$

where $K\geq 1$ . Also, by Lemma 3.8, for all $x,y\in f\mathbb {H}^n\subset \mathbb {H}^n$ , we have

(3.10) $$ \begin{align} h_{f\mathbb{H}^n}(f(x),f(y))= \rho_{f\mathbb{H}^n}(f(x),f(y)). \end{align} $$

The result now follows from (3.9), (3.10), and Lemma 3.8. The proof is now complete.

For $r\in (0,1)$ and $K\in [1,\infty )$ , the function $\varphi _K:[0,1]\rightarrow [0,1]$ is defined as follows:

$$ \begin{align*} \varphi_K(r)=\mu^{-1}\left(\frac{\mu(r)}{K}\right), \quad \varphi_K(0)=0; \varphi_K(1)=1, \end{align*} $$

where $\mu :(0,1)\rightarrow (0,\infty )$ is a decreasing homeomorphism given by

and F represents the Gaussian hypergeometric function. For additional information about the function $\varphi _K(r)$ and its approximation, readers are encouraged to consult [Reference Kargar, Rainio and Vuorinen17].

Theorem 3.11 If $f:\mathbb {B}^2\rightarrow \mathbb {B}^2$ is a nonconstant K-quasiregular mapping, then

$$ \begin{align*} h_{\mathbb{B}^2}(f(x),f(y))\leq c(K) \max\{h_{\mathbb{B}^2}(x,y), 2^{1-1/K}h_{\mathbb{B}^2}(x,y)^{1/K}\} \end{align*} $$

for all $x,y\in \mathbb {B}^2$ , where $c(K)=2\mathrm {arth}(\varphi _K(\mathrm {th}(1/2)))$ . In particular, $c(1)=1$ .

Proof Let $f:\mathbb {B}^2\rightarrow \mathbb {B}^2$ be a nonconstant K-quasiregular mapping. Then, by Theorem [Reference Hariri, Klén and Vuorinen10, Theorem 16.39], we have

(3.11) $$ \begin{align} \rho_{\mathbb{B}^2}(f(x),f(y))\leq c(K) \max\{\rho_{\mathbb{B}^2}(x,y), \rho_{\mathbb{B}^2}(x,y)^{1/K}\} \end{align} $$

for all $x,y\in \mathbb {B}^2$ , where $c(K)=2\mathrm {arth}(\varphi _K(\mathrm {th}(1/2)))$ . The desired assertion can be obtained by utilizing Lemma 3.7 and inequality (3.11).

4 Harmonic Schwarz lemma

This section first generalizes the Schwarz lemma for harmonic functions in the complex plane utilizing the Poisson integral formula. Then, it improves the Schwarz–Pick estimate for a real-valued harmonic function. First, we recall that the classical Schwarz lemma states that if $u:\mathbb {B}^2\rightarrow \mathbb {B}^2$ is a holomorphic function with $u(0)=0$ , then:

• • $|u(z)|\leq |z|$ for all $z\in \mathbb {B}^2$ ;

• • $|u'(0)|\leq 1$ .

Heinz (see [Reference Heinz12]) has obtained an improvement of the classical Schwarz lemma for a complex-valued harmonic function (see Lemma 4.1). A complex-valued function $f:G \to \mathbb {C}$ , where $f= u + i v$ is said to be harmonic if both $u:G \to \mathbb {R}$ and $v:G \to \mathbb {R}$ are harmonic in the sense defined above.

Lemma 4.1 Let $u:\mathbb {B}^2\rightarrow \mathbb {B}^2$ be a complex-valued harmonic function with $u(0)=0$ . Then

$$ \begin{align*} |u(z)|\leq \frac{4}{\pi}\arctan |z|. \end{align*} $$

The inequality is sharp for each point $z\in \mathbb {B}^2$ .

The following Theorem 4.2 is known as the Poisson integral formula (see, for example, [Reference Gwynne9]).

Theorem 4.2 Let u be a complex-valued function continuous on $\overline {B}^2(a,R)$ , $(R>0)$ , and harmonic on ${B}^2(a,R)$ . Then for $r\in [0,R)$ and $t\in \mathbb {R}$ the following formulas hold:

(4.1) $$ \begin{align} u\left(a+r e^{i t}\right)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{R^2-r^2}{R^2+r^2-2rR \cos(t-\theta)}u\left(a+R e^{i \theta}\right)\mathrm{d}\theta \end{align} $$

and

(4.2) $$ \begin{align} u(a)= \frac{1}{2\pi}\int_{-\pi}^{\pi}u\left(a+R e^{i \theta}\right)\mathrm{d}\theta. \end{align} $$

Motivated by Lemma 4.1 and applying Theorem 4.2, we derive the following Theorem 4.3 which is an extension of the above Schwarz lemma:

Theorem 4.3 Let $0< r<R$ and $M>0$ . If u is a complex-valued harmonic mapping in the disk $B^2(a,R)$ such that $|u(w)|\leq M$ for all $w\in B^2(a,R)$ , then

$$ \begin{align*} \left|u(a+z)-\frac{R^2-|z|^2}{R^2+|z|^2}u(a)\right|\leq\frac{2M}{\pi}\arctan\left(\frac{2R|z|}{R^2-|z|^2}\right),\quad z=re^{i t}. \end{align*} $$

The result is sharp.

Proof Suppose that $0<r<R$ . Applying formula (4.1) for $z=r$ , we obtain

$$ \begin{align*} &u\left(a+r\right)-\frac{R^2-r^2}{R^2+r^2}u(a)\\ &\quad=\frac{1}{2\pi}\int_{-\pi}^{\pi} \left(\frac{R^2-r^2}{R^2+r^2-2rR \cos(\theta)}-\frac{R^2-r^2}{R^2+r^2} \right)u\left(a+Re^{i\theta}\right)\mathrm{d}\theta\\ &\quad= \frac{rR(R^2-r^2)}{\pi(R^2+r^2)}\int_{-\pi}^{\pi}\frac{\cos(\theta)}{R^2+r^2-2rR \cos(\theta)} u\left(a+Re^{i\theta}\right)\mathrm{d}\theta. \end{align*} $$

By the last equality and the assumption $|u|\leq M$ , we obtain

(4.3) $$ \begin{align} \left|u\left(a+r\right)-\frac{R^2-r^2}{R^2+r^2}u(a)\right|\leq M\frac{rR(R^2-r^2)}{\pi(R^2+r^2)}\int_{-\pi}^{\pi}\frac{|\cos(\theta)|}{R^2+r^2-2rR \cos(\theta)} \mathrm{d}\theta. \end{align} $$

Now, we calculate the integral

$$ \begin{align*} I=\int_{-\pi}^{\pi}\frac{|\cos(\theta)|}{R^2+r^2-2rR \cos(\theta)} \mathrm{d}\theta. \end{align*} $$

It is easy to check that

$$ \begin{align*} I&=\int_{-\pi/2}^{\pi/2}\left(\frac{\cos(\theta)}{R^2+r^2-2rR \cos(\theta)}+\frac{\cos(\theta)}{R^2+r^2+2rR \cos(\theta)}\right) \mathrm{d}\theta\\ &=2(R^2+r^2)\int_{-\pi/2}^{\pi/2}\frac{\cos(\theta)}{(R^2+r^2)^2-4r^2 R^2 \cos^2(\theta)}\mathrm{d}\theta\\ &=4(R^2+r^2)\int_{0}^{\pi/2}\frac{\cos(\theta)}{(R^2-r^2)^2+4r^2 R^2 \sin^2(\theta)}\mathrm{d}\theta\\ &=\frac{2(R^2+r^2)}{rR(R^2-r^2)}\arctan\left(\frac{2rR}{R^2-r^2}\right). \end{align*} $$

Thus, from (4.3) follows that

$$ \begin{align*} \left|u\left(a+r\right)-\frac{R^2-r^2}{R^2+r^2}u(a)\right|\leq \frac{2M}{\pi}\arctan\left(\frac{2rR}{R^2-r^2}\right), \end{align*} $$

which implies the desired result. It is easy to see that the result is sharp for the function

$$ \begin{align*} u_0(z)=-\frac{2M}{\pi}\arg \left(\frac{R-z}{R+z}\right)=\frac{2M}{\pi}\arctan\left( \frac{2rR\sin \theta}{R^2-r^2}\right),\quad z=re^{it}, \end{align*} $$

or one of its rotations, where $0<r<R$ and $M>0$ , completing the proof.

Remark 4.4 It should be noted that Theorem 4.3 is also an extension of [Reference Pavlović21, Theorem 3.6.1]. Indeed, Pavlović proved that if $f:\mathbb {B}^2\rightarrow \overline {\mathbb {B}}^2$ is a complex-valued harmonic function, then the following sharp inequality holds:

$$ \begin{align*} \left|u(z)-\frac{1-|z|^2}{1+|z|^2}u(0)\right|\leq\frac{4}{\pi}\arctan |z|,\quad z=re^{i t}. \end{align*} $$

Let $\nabla u$ be the gradient of u at x defined by

$$ \begin{align*} \nabla u(x)=(\partial u/\partial x_1,\ldots,\partial u/\partial x_n). \end{align*} $$

In 1989 (see [Reference Colonna6]), Colonna proved the following Schwarz–Pick estimate for complex-valued harmonic functions u from the unit disk $\mathbb {B}^2$ to itself:

$$ \begin{align*} \left|\frac{\partial u(z)}{\partial z}\right|+\left|\frac{\partial u(z)}{\partial \overline{z}}\right|\leq \frac{4}{\pi}\frac{1}{1-|z|^2}, \quad z\in \mathbb{B}^2. \end{align*} $$

If u is a real-valued function, Kalaj and Vuorinen established the above Schwarz–Pick estimate as the following theorem; refer to [Reference Kalaj and Vuorinen16, Theorem 1.8] for details.

Theorem 4.5 Let u be a real harmonic function of the unit disk into $(-1, 1)$ . Then the following sharp inequality holds:

$$ \begin{align*} |\nabla u(z)|\leq \frac{4}{\pi}\frac{1-|u(z)|^2}{1-|z|^2}, \quad z\in \mathbb{B}^2. \end{align*} $$

In accordance with the findings of Chen [Reference Chen4, Theorem 1.2], the subsequent result has been derived:

Theorem 4.6 Let u be a real harmonic mapping of $\mathbb {B}^2$ into the open interval $(-1,1)$ . Then

$$ \begin{align*} |\nabla u(z)|\leq \frac{4}{\pi}\frac{\cos \left(\frac{\pi}{2}u(z)\right)}{1-|z|^2} \end{align*} $$

holds for $z\in \mathbb {B}^2$ . The inequality is sharp for any $z\in \mathbb {B}^2$ and any value of $u(z)$ , and the equality occurs for some point in $\mathbb {B}^2$ if and only if $u(z)=(4 \mathrm {Re}\{\arctan \, f(z)\})/\pi $ , $z\in \mathbb {B}^2$ with a Möbius transformation f of $\mathbb {B}^2$ onto itself.

In the subsequent discussion, we aim to expand upon Theorem 4.5 in the following manner: Furthermore, it is worth noting that our extension encompasses the findings presented in Theorem 6.26 of [Reference Axler, Bourdon and Ramey1].

Theorem 4.7 Let $\alpha $ and $\beta $ be two real numbers such that $\alpha <\beta $ . If $u:\mathbb {B}^2\rightarrow (\alpha ,\beta )$ is a real-valued harmonic function, then we have

$$ \begin{align*} |\nabla u(z)|\leq \frac{2(\beta-\alpha)}{\pi}\frac{1-\frac{4}{(\beta-\alpha)^2} \left|u(z)-\frac{\alpha+\beta}{2}\right|{}^2}{1-|z|^2},\quad z\in \mathbb{B}^2. \end{align*} $$

The result is sharp.

Proof Define $v(z)$ as

$$ \begin{align*} v(z)=\frac{2}{\beta-\alpha}\left(u(z)-\frac{\alpha+\beta}{2}\right),\quad z\in \mathbb{B}^2, \end{align*} $$

where $u:\mathbb {B}^2\rightarrow (\alpha ,\beta )$ is a real-valued harmonic function, $\alpha $ and $\beta $ are real numbers such that $\alpha <\beta $ . Then it is clear that v is a harmonic function of the unit disk $\mathbb {B}^2$ into $(-1,1)$ . Therefore, v satisfies the assumption of Theorem 4.5. Moreover, we have

$$ \begin{align*} \frac{2}{\beta-\alpha}|\nabla u|=|\nabla v|\leq \frac{4}{\pi}\frac{1-\frac{4}{(\beta-\alpha)^2} \left|u(z)-\frac{\alpha+\beta}{2}\right|{}^2}{1-|z|^2},\quad z\in \mathbb{B}^2, \end{align*} $$

which implies the desired result. To show that the result is sharp, we take the harmonic function

$$ \begin{align*} \ell(z)=\frac{\alpha+\beta}{2}+\frac{\beta-\alpha}{\pi}\arctan \frac{2y}{1-x^2-y^2},\quad z\in \mathbb{B}^2. \end{align*} $$

It is easy to see $\alpha <\ell (z)<\beta $ . A simple calculation yields

$$ \begin{align*} |\nabla \ell(0)|=\frac{2(\beta-\alpha)}{2}=\frac{2(\beta-\alpha)}{2}\cdot \frac{1-\frac{4}{(\beta-\alpha)^2}\left|\frac{\alpha+\beta}{2}-\frac{\alpha+\beta}{2} \right|{}^2}{1-0^2}, \end{align*} $$

which is the desired conclusion.

Applying Theorem 4.6, we get the following result:

Theorem 4.8 If $u:\mathbb {B}^2\rightarrow (\alpha ,\beta )$ is an into harmonic mapping, then

$$ \begin{align*} |\nabla u(z)|\leq \frac{2(\beta-\alpha)}{\pi}\frac{\cos \left(\frac{\pi}{\beta-\alpha}\left(u(z)-\frac{\alpha+\beta}{2}\right)\right)}{1-|z|^2}, \end{align*} $$

where $\alpha $ and $\beta $ are real numbers such that $\alpha <\beta $ . The result is sharp.

We conclude this paper by presenting the following open question:

Open question. What is the connection between the Harnack metric h and the hyperbolic metric $\rho $ in a simply connected Jordan domain in the complex plane $\mathbb {C}$ ?