On the Theorem of Wan for K-Quasiconformal Hyperbolic Harmonic Self Mappings of the Unit Disk

We give a new glance to the theorem of Wan (Theorem 1.1) which is related to the hyperbolic bi-Lipschicity of theK-quasiconformal, K > 1, hyperbolic harmonic mappings of the unit disk D onto itself. Especially, if f is such a mapping and f(0) = 0, we obtained that the following double inequality is valid 2|z|/(K + 1) 6 |f(z)| 6 √ K|z|, whenever z ∈ D.


Introduction
Suppose that ρ is a positive function defined and of the class C 2 in some subdomain (open and connected) Ω of the complex plane C and let z 0 ∈ Ω. Recall that the Gaussian curvature of the conformal metric ds 2 = ρ(z)|dz| 2 at the point z 0 is defined as (1) K ρ (z 0 ) = − 1 2 ( log ρ)(z 0 ) ρ(z 0 ) , where is the Laplace second order differential operator (the Laplacian).Also, the ρ-length of a rectifiable curve γ : [0, 1] → Ω is given by |γ| ρ = γ ρ(z)|dz|.Otherwise, the ρ-distance between the points z 1 and z 2 in Ω is defined as d ρ (z 1 , z 2 ) = inf |γ| ρ , where the infimum is taken over all rectifiable curves γ in Ω that join the points z 1 and z 2 .
Since, for arbitrary z ∈ D, we have for all z ∈ D. So, the conformal metric ds 2 = λ(z)|dz| 2 has the constant and negative Gaussian curvature on D. On the other hand, it is easy to verify that the corresponding distance function induced by this metric on D is given by the formula Definition 1.1.The hyperbolic metric on the unit disk is a conformal metric ds 2 = λ(z)|dz| 2 , where the density function λ is given by (2).The function d λ is called the hyperbolic distance on the unit disk D.
For further properties of the hyperbolic metric we refer to [2] and [9].
Let Ω and Ω be some subdomains of the complex plane C.
Definition 1.2.For a mapping f : Ω → Ω , which is of the class C 2 in Ω, we say that it is harmonic with respect to a conformal metric ds 2 = ρ(w)|dw| 2 defined on Ω (ρ is a positive function and of the class for all z ∈ Ω, where f z and f z are the partial derivatives of f in Ω related to the variables z and z, respectively.Here by f z z we denoted the second order partial derivative of the mapping It is obvious that in the presence of the Euclidean metric on the image subdomain Ω , i.e. in the case when the density function ρ ≡ 0 on Ω , the relation (4) defines a harmonic function, or Euclidean harmonic mapping, since ( f Note that if K = 1 the mapping f is a conformal mapping, since in that case f z ≡ 0 on Ω.By using a new approach and technique, in the article [5] we gave a new proof of Wan's result (see [12]) related to the bi-Lipschicity of the quasiconformal hyperbolic harmonic diffeomorphisms of the unit disk.More specifically, we constructed some conformal metrics on the unit disk D and, by understanding their properties and by calculating their Gaussian curvatures, we applied some versions of the results that are of the Ahlfors-Schwarz-Pick type to show that every K-quasiconformal mapping f of the unit disk D onto itself, which is also harmonic with respect to the hyperbolic metric ds 2 = λ(w)|dw| 2 on D, is a quasi-isometry with respect to the hyperbolic metric.Moreover, such a mapping f is (2/(K + 1), √ K) bi-Lipschitz with respect to the hyperbolic metric, too.
Theorem 1.1 (Wan [12], KM [5]).Let f ∈ C 2 (D) be a K-quasiconformal mapping of the unit disk D onto itself which is harmonic with respect to the hyperbolic metric ds 2 = λ(w)|dw| 2 on D. Then f is a (2/(K + 1), √ K) bi-Lipschitz with respect to the hyperbolic metric.Since √ K (K + 1)/2, then f is also a quasi-isometry with respect to the hyperbolic metric.

The main result
Suppose now that a given mapping f satisfies the conditions of the previous theorem.In addition, suppose that f (0) = 0.According to the Theorem 1.1, we have for all z ∈ D, and since d λ (r, 0) = ln 1 + r 1 − r , for all 0 r < 1, we get , for all z ∈ D.
To obtain the main result of this paper, we have to prove the following lemma.
Proof.For the defined function a we easily get for all 0 < x < 1.On the other hand, for its second derivative we have, Therefore, since for α > 1, we obtain that in this case the function a is concave on (0, 1).Otherwise, if and the function a is then convex on (0, 1).Now, the statement easily follows from the fact that a + (0) = α, where a + (0) is the right derivative of the function a at the point x = 0.
We are ready now to prove the main result.
Theorem 2.1.Let f ∈ C 2 (D) be a K-quasiconformal mapping of the unit disk D onto itself which is harmonic with respect to the hyperbolic metric ds 2 = λ(w)|dw| 2 on D. Suppose, in addition, that f (0) = 0.Then, for all z ∈ D we have for all z ∈ D.
Proof.The proof is a trivial consequence of the inequalities ( 5) and ( 6), and of the Lemma 2.1.
Remark 2.1.In [5] we obtained some version of the Theorem 1.1 that are related to the K-quasiconformal harmonic mappings f , which are harmonic with respect to some conformal metric defined on the image subdomain, and with the property that its Gaussian curvature is not greater then some negative constant −a, a > 0. Therefore, we could easily generalize the Theorem 2.1 in this case.

Example 1 . 1 .
Let D = {z ∈ C : |z| < 1} be the unit disk in C. Consider a conformal metric ds 2 = λ(z)|dz| 2 on D, where the corresponding density function λ is defined as