1 Introduction and Main Results

Martio [22] was the first who considered the study on harmonic quasiconformal mappings in \({\mathbb C}\). In the recent years the articles [8, 14, 16,17,18, 26] brought much light on this topic. In [6, 21], the Lipschitz characteristic of \((K, K^{\prime })\)-quasiconformal mappings has been discussed. In [20], the authors proved that a K-quasiconformal harmonic mapping from the unit disk \({\mathbb D}\) onto itself is bi-Lipschitz with respect to hyperbolic metric, and also proved that a K-quasiconformal harmonic mapping from the upper half-plane \(\mathbb {H}\) onto itself is bi-Lipschitz with respect to hyperbolic metric. In [23], the authors proved that a K-quasiconformal harmonic mapping from D to \(D'\) is bi-Lipschitz with respect to quasihyperbolic metrics on D and \(D'\), where D and \(D'\) are proper domains in \({\mathbb C}\). Important definitions will be included later in this section.

In [15], Kalaj considered the bi-Lipschitz continuity of K-quasiconformal solution of the inequality

$$\begin{aligned} |\Delta f|\le B|D f|^2. \end{aligned}$$
(1.1)

Here \(\Delta f\) represents the two-dimensional Laplacian of f defined by \(\Delta f=f_{xx}+f_{yy}=4f_{z\overline{z}} \) and the mapping f satisfying the Laplace equation \(\Delta f=0\) is called harmonic. For \(z=x+iy\) and \(f=u+iv,\) Df denotes the Jacobian matrix

$$\begin{aligned} \left( \begin{array}{cc} u_x &{}\quad u_y \\ v_x &{}\quad v_y \\ \end{array} \right) \end{aligned}$$

so that \(J_f=|f_z|^2-|f_{\overline{z}}|^2\) is the Jacobian of f.

The first aim of this paper is to consider the Lipschitz continuity of \((K, K')\)-quasiconformal solution of the inequality (1.1) with respect to the distance ratio metric.

Theorem 1.4

Let f be a \((K, K')\)-quasiconformal \(C^2\) mapping from the unit disk \(\mathbb {D}=\{z:\,|z|<1\}\) onto itself, satisfying the inequality (1.1) and \(f(0)=0\). Then f is Lipschitz continuous with respect to the distance ratio metric.

The proof of Theorem 1.1 will be presented in Sect. 2. Before we proceed further, let us fix up further notation, preliminaries and remarks.

1.1 \((K, K^{\prime })\)-Quasiconformal Mappings

We say that a function \(u:\, D\rightarrow \mathbb {R}\) is absolutely continuous on lines, ACL in brief, in the domain D if for every closed rectangle \(R\subset D\) with sides parallel to the axes x and y, u is absolutely continuous on almost every horizontal line segment and almost every vertical line segment in R. Such a function has, of course, partial derivatives \(u_x\) and \(u_y\) a.e. in D (cf. [1]). Further, we say \(u\in ACL^{2}\) if \(u\in ACL\) and its partial derivatives are locally \(L^2\) integrable in D.

A sense-preserving continuous mapping \(f:\, D\rightarrow \Omega \) is said to be

  1. 1.

    \((K, K^{\prime })\) -quasiregular if f is \(ACL^{2}\) in D, \(J_{f}\not =0\) a.e. in D and there are constants \(K\ge 1\) and \(K'\ge 0\) such that \(|D f|^2\le KJ_f+K'\) a.e. in D, where \(|D f|=|f_z|+|f_{\overline{z}}|\);

  2. 2.

    K -quasiregular if \(K'=0\).

In particular, f is called \((K, K')\) -quasiconformal if f is a \((K, K')\)-quasiregular homeomorphism; and f is K-quasiconformal if f is a K-quasiregular homeomorphism.

Here are some basic comments on these mappings. From [6, Example 2.1] and [21, Example 2.1] we know that there are \((K, K')\)-quasiregular mappings which are not \(K_1\)-quasiregular for any \(K_1\ge 1\). Moreover, it is known that (see [6, Example 4.1]) there are \((K, K')\)-quasiconformal mappings whose inverses are not \((K_1, K_1')\)-quasiconformal for any \(K_1\ge 1\) and \(K_1'\ge 0\).

Remark 1.2

If f is a \((K, K')\)-quasiregular mapping, g is a analytic function and \(|g'|\) is bounded by a constant L, then \(f\circ g\) is \((K, K'L^2)\)-quasiregular mapping.

A mapping \(f:\,D\rightarrow \Omega \) is proper if the preimage of every compact set in \(\Omega \) is compact in D (cf. [19, p. 4051] or [30, p. 17]).

1.2 The Distance Ratio Metric

For a subdomain \(G\subset \mathbb {C}\) and for all z, \(w\in G\), the distance ratio metric \(j_{G}\) is defined as

$$\begin{aligned} j_{G}(z,w)=\log \left( 1+\frac{|z-w|}{\min \{\delta _G(z),\delta _G(w)\}}\right) , \end{aligned}$$

where \(\delta _G(z)\) denotes the Euclidean distance from z to \(\partial G\). The distance ratio metric was introduced by Gehring and Palka [12] and in the above simplified form by Vuorinen [31]. However, the distance ratio metric \(j_{G}\) is not invariant under Möbius transformations. Therefore, it is natural to consider the Lipschitz continuity of conformal mappings or Möbius transformations with respect to the distance ratio metric. Gehring and Osgood [11] proved that the distance ratio metric is not altered by more than a factor of 2 under Möbius transformations.

Theorem A

([11, Proof of Theorem 4]) If G and \(G'\) are proper subdomains of \(\mathbb {R}^{n}\) and if f is a Möbius transformation of G onto \(G'\), then \(j_{G'}(f(x),f(y))\le 2j_{G}(x,y)\) for all x, \(y\in G\).

Recall that a mapping \(f:\,D\rightarrow \Omega \) is said to be Lipschitz continuous (resp. Lipschitz continuous with respect to the distance ratio metric) if there exists a positive constant \(L_{1}\) (resp. a positive constant L) such that for all \(z, w\in D\),

$$\begin{aligned} |f(z)-f(w)|\le L_{1}|z-w| \quad \text{(resp. } j_{\Omega }(f(z),f(w))\le L j_{D}(z,w)). \end{aligned}$$

In 2011, Kalaj and Mateljević [17] proved that every quasiconformal \(C^2\) diffeomorphism f from the domain \(\Omega \) with \(C^{1, \alpha }\) compact boundary onto the domain G with \(C^{2, \alpha }\) compact boundary satisfying the Poisson differential inequality

$$\begin{aligned} |\Delta f|\le B|D f|^2+C \end{aligned}$$
(1.2)

for some constants \(B\ge 0\) and \(C\ge 0\), is Lipschitz continuous respect to Euclidean metric. Clearly if \(B=C=0\) in (1.2), then f is harmonic.

Recently, the authors in [6, Theorem 1.1] proved the following theorem:

Theorem B

Suppose f is a proper \((K, K')\)-quasiregular \(C^2\) mapping of a Jordan domain D with \(C^{1,\alpha }\) boundary onto a Jordan domain \(\Omega \) with \(C^{2,\alpha } \) boundary. If f satisfies the partial differential inequality (1.2) for constants \(B>0\) and \(C\ge 0\), then f has bounded partial derivatives in D. In particular, f is Lipschitz continuous.

Remark 1.3

From Theorem B we infer that if \(f:\,{\mathbb D}\rightarrow {\mathbb D}\) satisfies the conditions of Theorem 1.1, then there exists a constant M such that \(|D f|\le M\). Hence for all \(z, w\in {\mathbb D}\), we have \(|f(z)-f(w)|\le M|z-w|\) and \(|f(z)|\le M|z|\). If \(M <1\), we get

$$\begin{aligned} j_{{\mathbb D}}(f(z), f(w))= & {} \log \left( 1+\frac{|f(z)-f(w)|}{\min \{\delta _{{\mathbb D}}(f(z)),\delta _{{\mathbb D}}(f(w))\}}\right) \\\le & {} \log \left( 1+\frac{M|z-w|}{M\,\min \{\delta _{{\mathbb D}}(z),\delta _{{\mathbb D}}(w)\}}\right) \\\le & {} j_{{\mathbb D}}(z, w), \end{aligned}$$

which clearly shows that \(f:\,{\mathbb D}\rightarrow {\mathbb D}\) is a Lipschitz continuous function with respect to the distance ratio metric. \(\Box \)

In order to state our next result, we need to recall the definition of hypergeometric series. For \(a, b, c\in {\mathbb R}\) with \(c\ne 0, -1, -2, \ldots \), the hypergeometric function is defined by the power series

$$\begin{aligned} F(a, b; c; z):={}_2F_1(a,b;c;z)=\sum _{n=0}^{\infty }\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!},\;\;|z|<1, \end{aligned}$$

where \((a)_0=1\) and \((a)_n=a(a+1)\cdots (a+n-1)\) for \(n=1, 2, \ldots \) are the Pochhammer symbols. Obviously, for \(n=0, 1, 2, \ldots \), \((a)_n=\Gamma (a+n)/\Gamma (a)\). In particular, for \(a, b, c>0\) and \(a+b<c\), we have (cf. [3, 4])

$$\begin{aligned} F(a, b; c; 1)=\lim _{z\rightarrow 1^{-}}F(a, b; c; z)=\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}<\infty . \end{aligned}$$

Consider the operator equation

$$\begin{aligned} T_{\alpha }(f)=0 ~ \text{ in } {\mathbb D}, \end{aligned}$$
(1.3)

where \(f:\,{\mathbb D}\rightarrow {\mathbb C}\), \(\alpha \in {\mathbb R}\), and

$$\begin{aligned} T_\alpha =-\frac{\alpha ^2}{4}(1-|z|^2)^{-\alpha -1} +\frac{\alpha }{2}(1-|z|^2)^{-\alpha -1}\left( z\frac{\partial }{\partial z}+\overline{z}\frac{\partial }{\partial \overline{z}}\right) +(1-|z|^2)^{-\alpha }\frac{\partial ^2}{\partial z\partial \overline{z}} \end{aligned}$$

is the second order elliptic partial differential operator defined on the unit disk \({\mathbb D}\). In the case of \(\alpha =0\), \(T_{\alpha }(f)=0\) is equivalent to saying that f is harmonic in \({\mathbb D}\). More generally, if f satisfies (1.3) with \(\alpha =2(n-1)\), then f is polyharmonic (or n-harmonic) in \({\mathbb D}\), where \(n\in \{1, 2, \ldots \}\) (cf. [2, 5, 9, 27]). Recently, several new properties of polyharmonic mappings are discussed in [2]. The following result concerns the solutions to the equation (1.3).

Lemma C

[25, Theorem 2.2] Let \(\alpha \in {\mathbb R}\) and \(f\in C^2({\mathbb D})\). Then f satisfies (1.3) if and only if it has a series expansion for \(z\in {\mathbb D}\) of the form

$$\begin{aligned} f(z)=\sum _{k=0}^{\infty }c_kF\left( -\frac{\alpha }{2}, k-\frac{\alpha }{2}; k+1; |z|^2\right) z^k+\sum _{k=1}^{\infty }c_{-k}F\left( -\frac{\alpha }{2}, k-\frac{\alpha }{2}; k+1; |z|^2\right) {\overline{z}}^k,\nonumber \\ \end{aligned}$$
(1.4)

where \(\{c_k\}_{k=-\infty }^\infty \) is a sequence of complex numbers satisfying

$$\begin{aligned} \limsup _{|k|\rightarrow \infty }|c_k|^{\frac{1}{|k|}}\le 1. \end{aligned}$$
(1.5)

In particular, the expansion (1.4), subject to (1.5), converges in \(C^\infty ({\mathbb D})\), and every solution f of (1.3) is \(C^\infty \)-smooth in \({\mathbb D}\).

In [10, 24], the authors gave some properties of solution to (1.3) whereas in [28, 29], the authors considered the Lipschitz continuity of the distance-ratio metric under some Möbius automorphisms of the unit ball and conformal mappings from \(\mathbb {D}\) to \(\mathbb {D}\). In [7], the authors discussed the Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric. Thus, it is natural to investigate Lipschitz continuity of the solution of (1.3) in \({\mathbb D}\) with respect to the distance ratio metric. We now state our next result.

Theorem 1.4 For \(\alpha >-1\), let \(f:\, {\mathbb D}\rightarrow {\mathbb D}\) be a \(C^2\)-solution to (1.3) with the series expansion of the form (1.4) and \(f(0)=0\). If

$$\begin{aligned} \sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}\le 1, \end{aligned}$$
(1.6)

then

$$\begin{aligned}j_{\mathbb {D}}(f(z),f(w))\le j_{\mathbb {D}}(z,w),\end{aligned}$$

and this inequality is sharp. That is, f is Lipschitz continuous with respect to the distance ratio metric.

Remark 1.5

In Theorem 1.4, we restrict \(\alpha >-1\), see [25, proposition 1.4] for the reason for this constraint.

The proof of Theorem 1.4 will be presented in Sect. 2.

2 Proof of Theorems 1.1 and 1.4

First we shall deal with the Lipschitz continuity of certain \((K, K')\)-quasiconformal mappings and then consider the Lipschitz continuity of the solution to the differential operator \(T_\alpha \) with respect to the distance ratio metric.

Lemma 2.1

Assume the hypotheses of Theorem 1.1. Then there exists a constant \(C(K, K', B)\) such that for \(z\in {\mathbb D}\)

$$\begin{aligned} \frac{1-|z|^2}{1-|f(z)|^2}\le C(B,K, K'). \end{aligned}$$
(2.1)

Proof

By assumption f is a \((K, K')\)-quasiconformal \(C^2\) mapping from \({\mathbb D}\) onto itself, satisfying the inequality (1.1) and \(f(0)=0\). For convenience, we denote the class of all such functions f by \(\mathcal {QC}({\mathbb D}, B, K, K')\). Then there is a positive constant A not depending on f such that the function \(\varphi _f\), \(f\in \mathcal {QC}({\mathbb D}, B, K, K')\), defined by

$$\begin{aligned} \varphi _f(z)=-\frac{1}{A}+\frac{1}{A}e^{A(|f(z)|-1)} \end{aligned}$$

is subharmonic in \({\mathbb D}\).

Now, let us prove the existence of such an A. Take

$$\begin{aligned} \psi (\rho )=-\frac{1}{A}+\frac{1}{A}e^{A(\rho -1)}. \end{aligned}$$

Then \(\psi '(\rho )=e^{A(\rho -1)}\) and \(\psi ''(\rho )=Ae^{A(\rho -1)}\). On the other hand, using \(f_z=(1/2)(f_x-if_y)\) and \(f_{\overline{z}} =(1/2)(f_x+if_y)\), we find that

$$\begin{aligned} |D |f||^2=|f|_x^2+|f|_y^2=(|f|_z+|f|_{\overline{z}})^2+i^2(|f|_z-|f|_{\overline{z}})^2=4|f|_z|f|_{\overline{z}}, \end{aligned}$$

and thus,

$$\begin{aligned} \Delta \varphi _f=\psi ''(|f|)|D |f||^2+\psi '(|f|)\Delta |f|. \end{aligned}$$
(2.2)

Furthermore, put \(s=f/|f|\). By elementary calculations we see that the following equalities hold:

$$\begin{aligned} \Delta |f|=\frac{|f_z|^2+|f_{\overline{z}}|^2}{|f|}-2\mathrm{Re}\left( \overline{f}^{\frac{1}{2}}f^{-\frac{3}{2}}f_zf_{\overline{z}}\right) +\mathrm{Re}\,(\overline{s}\Delta f) \end{aligned}$$

and

$$\begin{aligned} |s_z|^2=|s_{\overline{z}}|^2 =\frac{1}{4}\frac{|f_z|^2+|f_{\overline{z}}|^2}{|f|^2}-\frac{1}{2|f|}\mathrm{Re}\left( \overline{f}^{\frac{1}{2}}f^{-\frac{3}{2}}f_zf_{\overline{z}}\right) . \end{aligned}$$

Then we know that

$$\begin{aligned} \Delta |f|=|f|\cdot |D s|^2+\mathrm{Re}\,(\overline{s}\Delta f). \end{aligned}$$
(2.3)

We continue the discussion by setting \(\rho =|f|\). According to [6, Lemma 3.1], we have

$$\begin{aligned} |D \rho |\ge \frac{|D f|}{K}-\frac{\sqrt{K'}}{K}, \end{aligned}$$
(2.4)

Using (1.1), (2.2), (2.3) and (2.4), it follows finally that

$$\begin{aligned} \Delta \varphi _f=&\,e^{A(\rho -1)}\Big [A|D \rho |^2+\rho |D s|^2+\mathrm{Re}\,(\overline{s}\Delta f)\Big ]\\ \ge&\, e^{A(\rho -1)}\Big [\frac{A}{K^2}(|D f|-\sqrt{K'})^2-B|D f|^2\Big ]\\ =&\,e^{A(\rho -1)}\Big (\frac{A-BK^2}{K^2}|D f|^2-\frac{2A\sqrt{K'}}{K^2}|D f|+\frac{AK'}{K^2}\Big ). \end{aligned}$$

We obtain from Theorem B that f is Lipschitz continuous, and then there exists a constant M such that \(|D f|\le M\). Hence, if we choose an appropriate A, satisfying \(A\ne BK^2\) and

$$\begin{aligned} \frac{A\sqrt{K'}-\sqrt{ABK'K^2}}{A-BK^2}\ge M, \end{aligned}$$

i.e.

$$\begin{aligned} (\sqrt{K'}-M)^2A^2+[2BMK^2(\sqrt{K'}-M)-BK^2K']A+B^2M^2K^4\ge 0, \end{aligned}$$
(2.5)

we obtain the inequality \(\Delta \varphi _f(z)\ge 0\) for \(|z|<1\). We next show that this choice of A is possible.

If \(M=\sqrt{K'}\), then there exists an appropriate value of A satisfying the inequality (2.5). If \(M\ne \sqrt{K'}\) and \(K'+4M^2-4M\sqrt{K'}\le 0\), then (2.5) holds for all A. If \(M\ne \sqrt{K'}\) and \(K'+4M^2-4M\sqrt{K'}>0\), then (2.5) holds for all

$$\begin{aligned} A\ge \frac{BK^2K'-2BMK^2(\sqrt{K'}-M)+BK^2\sqrt{K'(K'+4M^2-4M\sqrt{K'})}}{2(\sqrt{K'}-M)^2}. \end{aligned}$$

In conclusion, there must exist an appropriate A such that \(\Delta \varphi _f(z)\ge 0\) for \(|z|<1\).

Define

$$\begin{aligned} F(z)=\sup \{\varphi _f(z):\, f\in \mathcal {QC}({\mathbb D}, B, K, K')\}. \end{aligned}$$

We prove that F is subharmonic in \({\mathbb D}\). By [13, Theorem 1.6.2], we only need to prove that F is continuous. Define \(h(z)=e^{A(|z|-1)}, \;|z|<1.\) Elementary calculations show that

$$\begin{aligned} h_z(z)=\frac{A}{2}\frac{\overline{z}}{|z|}e^{A(|z|-1)} \quad \text{ and }\quad h_{\overline{z}}(z)=\frac{A}{2}\frac{z}{|z|}e^{A(|z|-1)}. \end{aligned}$$

Then \(|D h|=|h_z|+|h_{\overline{z}}|=Ae^{A(|z|-1)}<A\) which implies that

$$\begin{aligned} |h(z)-h(z')|\le A|z-z'| \quad \text{ for } z, z'\in {\mathbb D}. \end{aligned}$$

According to Theorem B, we know that f is Lipschitz continuous. Therefore

$$\begin{aligned} |\varphi _f(z)-\varphi _f(z')|= \frac{1}{A}\left| e^{A(|f(z)|-1)}-e^{A(|f(z')|-1)}\right| \le |f(z)-f(z')| \le M|z-z'|, \end{aligned}$$

where M is a constant. Hence, \(|F(z)-F(z')|\le M|z-z'|\) so that F is continuous. Finally, from the similar proof of [15, Lemma 2.3], we complete the proof. \(\square \)

2.1 Proof of Theorem 1.1

From the hypotheses of Theorem 1.1 and Lemma 2.1, we obtain that

$$\begin{aligned} \frac{1-|z|^2}{1-|f(z)|^2}\le C(K, K', B) \end{aligned}$$

and thus, we obtain that

$$\begin{aligned} \frac{1-|z|}{1-|f(z)|}\le C(K, K', B)\frac{1+|f(z)|}{1+|z|}\le 2C(K, K', B). \end{aligned}$$

Moreover, from Theorem B, we see that f is Lipschitz continuous and therefore, there exists a constant \(M_1\) such that \(|D f|\le M_1\). Now, we choose an appropriate constant M satisfying \(M>\max \{M_1, 1/(2C(K, K', B))\}\) so that \(|D f|\le M\). Consequently, using the Bernoulli inequality, for any two points z and w in \({\mathbb D}\), we have

$$\begin{aligned} j_{{\mathbb D}}(f(z), f(w))= & {} \log \left( 1+\frac{|f(z)-f(w)|}{\min \{\delta _{{\mathbb D}}(f(z)),\delta _{{\mathbb D}}(f(w))\}}\right) \\\le & {} \log \left( 1+2C(K, K', B)M\frac{|z-w|}{\min \{\delta _{{\mathbb D}}(z),\delta _{{\mathbb D}}(w)\}}\right) \\\le & {} 2C(K, K', B)Mj_{{\mathbb D}}(z, w) \end{aligned}$$

and thus, the proof of the theorem is complete. \(\square \)

2.2 Proof of Theorem 1.4

For convenience, let \(g(t)=F(-\frac{\alpha }{2}, k-\frac{\alpha }{2}; k+1; t). \) For \(z,w\in \mathbb {D}\), let us assume that \(|f(z)|\ge |f(w)|\). Then

$$\begin{aligned}&|f(z)-f(w)|\\&\quad =\left| \sum _{k=1}^{\infty }c_{k}(|g(|z|^2)z^k-g(|w|^2)w^k)+\sum _{k=1}^{\infty }c_{-k}(|g(|z|^2){\overline{z}}^k-g(|w|^2){\overline{w}}^k)\right| \\&\quad \le |z-w|\sum _{k=1}^{\infty }\frac{|g(|z|^2)z^k-g(|z|^2)w^k+g(|z|^2)w^k-g(|w|^2)w^k|}{|z-w|}|c_{k}|\\&\qquad +|z-w|\sum _{k=1}^{\infty }\frac{|g(|z|^2){\overline{z}}^k-g(|z|^2){\overline{w}}^k+g(|z|^2){\overline{w}}^k-g(|w|^2){\overline{w}}^k|}{|z-w|}|c_{-k}|\\&\quad \le |z-w|\sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\left( \sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}\right. \\&\qquad \left. \times |z|^{2n}(|z|^{k-1}+|z|^{k-2}|w|+\cdots +|w|^{k-1})\right. \\&\qquad \left. +\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}|w|^k(|z|^{2n-1}+|z|^{2n-2}|w|+\cdots +|w|^{2n-1})\right) \\&\quad \le |z-w|\sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}\sum _{s=0}^{2n+k-1}|z|^s, \end{aligned}$$

and

$$\begin{aligned} 1-|f(z)|\ge & {} \sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}-|f(z)|\\\ge & {} \sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\Big (\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}-g(|z|^2)|z|^k\Big )\\= & {} \sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}(1-|z|^{2n+k})\\= & {} (1-|z|)\sum _{k=1}^{\infty }(|c_{k}|+|c_{-k}|)\sum _{n=0}^{\infty }\frac{(-\frac{\alpha }{2})_n(k-\frac{\alpha }{2})_n}{(k+1)_n n!}\sum _{s=0}^{2n+k-1}|z|^s, \end{aligned}$$

so that, using the Bernoulli inequality, we have

$$\begin{aligned} j_{\mathbb {D}}(f(z),f(w))= & {} \log \left( 1+\frac{|f(z)-f(w)|}{1-|f(z)|}\right) \\\le & {} \log \left( 1+\frac{|z-w|}{1-|z|}\right) \\\le & {} j_{\mathbb {D}}(z,w). \end{aligned}$$

As in [7, Theorem 7], the mapping \(f(z)= |z|^{2(p-1)}z^{m}\) or \(f(z)= |z|^{2(p-1)}\overline{z}^{m}\) for p, \(m\ge 1\), shows the sharpness in the last inequality. The proof of the theorem is complete. \(\Box \)