On Mori's theorem for quasiconformal maps in the $n$-space

R. Fehlmann and M. Vuorinen proved in 1988 that Mori's constant $M(n,K)$ for $K$-quasiconformal maps of the unit ball in $\mathbf{R}^n$ onto itself keeping the origin fixed satisfies $M(n,K) \to 1$ when $K\to 1 .$ We give here an alternative proof of this fact, with a quantitative upper bound for the constant in terms of elementary functions. Our proof is based on a refinement of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an explicit version of the Schwarz lemma for quasiconformal self-maps of the unit disk. Some experimental results are provided to compare the various bounds for the Mori constant when $n=2 .$


Introduction
Distortion theory of quasiconformal and quasiregular mappings in the Euclidean n-space R n deals with estimates for the modulus of continuity and change of distances under these mappings. Some of the examples are the Hölder continuity, the quasiconformal counterpart of the Schwarz lemma, and Mori's theorem. The investigation of these topics started in the early 1950's for the case n = 2 and ten years later for the case n ≥ 3 . Many authors have contributed to the distortion theory, for some historical remarks see [Vu1,11.50].
As in [FV] we define Mori's constant M(n, K) in the following way. Let QC K , K ≥ 1, stand for the family of all K-quasiconformal maps of the unit ball B n onto itself keeping the origin pointwise fixed. Note that it is a well-known basic fact that an element in the set QC K can be extended by reflection to a K-quasiconformal map of the whole space R n = R n ∪ {∞} onto itself keeping the point ∞ fixed. Then for all K ≥ 1, n ≥ 2 , there exists a least constant M(n, K) ≥ 1 such that for all f ∈ QC K , x, y ∈ B n .
L. V. Ahlfors [A1] proved in 1954 that M(2, K) ≤ 12 K 2 and this property was refined by A. Mori [Mo] in 1956 to the effect that M(2, K) ≤ 16 and 16 cannot be replaced by a smaller constant independent of K . This result can also be found in [A2], [FM], and [LV]. On the other hand the trivial observation that 16 fails to be a sharp constant for K = 1 led to the following conjecture, which is still open in 2009.
1.2. The Mori Conjecture. M(2, K) = 16 1−1/K . O. Lehto and K.I. Virtanen demonstrated in 1973 [LV,pp. 68] that M(2, K) ≥ 16 1−1/K (this lower bound was not given in the 1965 German edition of the book). It is natural to expect that for a fixed n ≥ 2, M(n, K) → 1 when K → 1 and this convergence result with an explicit upper bound for M(n, K) was proved by R. Fehlmann and M. Vuorinen [FV]. A counterpart of this result for the chordal metric was proved recently by P. Hästö in [H].
1.3. Theorem. [FV,Theorem 1.3] Let f be a K-quasiconformal mapping of B n onto B n , n ≥ 2, f (0) = 0. Then for all x, y ∈ B n where α = K 1/(1−n) and the constant M(n, K) has the following three properties: (1) M(n, K) → 1 as K → 1, uniformly in n , (2) M(n, K) remains bounded for fixed K and varying n , (3) M(n, K) remains bounded for fixed n and varying K .
For n = 2 , the first majorants with the convergence property in 1.3(1) were proved only in the mid 1980s and for n ≥ 3 in [FV]. In [FV] a survey of the various known bounds for M(n, K) when n ≥ 2 can be found -that survey reflects what was known at the time of publication of [FV]. Some earlier results on Hölder continuity had been proved in [G], [MRV], [R], [S].
Step by step the bound for Mori's constant was reduced during the past twenty years. As far as we know, the best upper bound known today for n = 2 is M(2, K) ≤ 46 1−1/K due to S.-L. Qiu [Q] (1997). Refining the parallel work [FV], G.D. Anderson and M. K. Vamanamurthy proved the following theorem in [AV].
The first main result of this paper is Theorem 1.6 which improves on Theorem 1.5.
(2) There exists a number K 1 > 1 such that for all K ∈ (1, K 1 ) the function h has a minimum at a point t 1 with t 1 > 1 and Moreover, for β ∈ (1, 2) we have In particular, h(t 1 ) → 1 when K → 1 .
The last statement shows that Theorem 1.6 is better than the result of Anderson and Vamanamurthy, Theorem 1.5, at least for values of K close to the critical value 1, because the constant of Theorem 1.5 satisfies 4λ The main method of our proof is to replace the argument of Anderson and Vamanamurthy by a more refined inequality from [Vu2] and to introduce an additional parameter (t in the above theorem) which will be chosen in an optimal way. The fact that this refined inequality is essentially sharp for values of t large enough, was recently proved by V. Heikkala and M. Vuorinen in [HV]. This gave us a hint that the inequality from [Vu2] might lead to an improvement of the results in [AV]. For the case n = 2 a numerical comparison of our bound (1.8) to Mori's conjectured bound, to the bound in Theorem 1.5 and to the bound in [FV] is presented in tabular and graphical form at the end of the paper.
We conclude this paper by discussing the Schwarz lemma for plane quasiconformal self-mappings of the unit disk, formulated in terms of the hyperbolic metric. The long history of this result is summarized in [Vu1,p.152,11.50]. An up-to-date form of the Schwarz lemma was given in [Vu1,Theorem 11.2] and it will be stated for convenient reference also below as Theorem 4.4. A particular case, formula (4.6), was rediscovered by D.B.A. Epstein, A. Marden and V. Markovic [EMM,Thm 5.1].
We use the notations ch, th, arch and arth as in [Vu1], to denote the hyperbolic cosine, tangent and their inverse functions, resp. The second main result of this paper is an explicit form of the Schwarz lemma for quasiregular mappings, Theorem 1.10. We believe that in this simple form the result is new and perhaps of independent interest. The constant c(K) below involves the transcendental function ϕ K defined in Section 4.

The main results
We shall follow here the standard notation and terminology for K-quasiconformal and K-quasiregular mappings in the Euclidean n-space R n , see e.g. [V], [Vu1], and we also recall some basic notation. For the modulus M(Γ) of a curve family Γ and its basic properties see [V] and [Vu1].
Let D and D ′ be domains in R n , K ≥ 1, and let f : for every curve family Γ in D [V].
For subsets E, F, D ⊂ R n we denote by ∆(E, F ; D) the family of all curves joining E and F in D. For brevity we write ∆(E, F ) = ∆(E, F ; R n ) . A ring is a domain in R n , whose complement consists of two compact and connected sets. If these sets are E and F , then the ring is denoted by R(E, F ) . The capacity of a ring The complementary components of the Grötzsch ring R G,n (s) are B n and [se 1 , ∞], s > 1, while those of the Teichmüller ring R T,n (t) are [−e 1 , 0] and [te 1 , ∞], t > 0. The conformal capacities of R G,n (s) and R T,n (t) are denoted by respectively. Here γ n : (1, ∞) → (0, ∞) and τ n : (0, ∞) → (0, ∞) are decreasing homeomorphisms and they satisfy the fundamental identity (2.1) γ n (s) = 2 n−1 τ n (s 2 − 1), t > 1 , see e.g. [Vu1,5.53].
Proof. It is well-known that the above definition defines g as a K-quasiconformal homeomorphism. The formula (2.6) is well-known (see [AVV2,Theorem 4.2]) and (2.7) follows easily.
We consider Teichmüller's extremal problem, which will be used to provide a key estimate in what follows. For x ∈ R n \ {0, e 1 }, n ≥ 2, define where the infimum is taken over all the pairs of continua E and F in R n with 0, e 1 ∈ E and x, ∞ ∈ F . Note that Lemma 2.8 gives the lower bound for p n (x) in Lemma 2.9.
2.11. Notation. For t > 0, x, y ∈ B n , we write By the triangle inequality we have Proof. Let Γ be the family ∆(E, F ) and let E and F be connected sets as in Lemma 2.9 with x, y ∈ E, z, ∞ ∈ F , where z = −tx/|x| and Γ ′ = f (Γ). By Lemma 2.8 and (2.10), we have The basic identity (2.1) yields Applying γ −1 n to (2.14) we have Because f B n ⊂ B n , by (2.6) and (2.4) we know that by inequalities (2.2) and (2.3). Exchanging the roles of x and y we see that Setting t = 1, we get the following corollary.
Proof. The proof is similar to the above proof except that here we consider the particular case t = 1. Because f B n ⊂ B n , we know that |f (2.2) and (2.3). Exchanging the roles of x and y we get 2.17. Corollary. For n ≥ 2, K ≥ 1, t ≥ 1, let f be as in Theorem 2.13. Then for all x, y ∈ B n , and Proof. Inequality (2.18) follows because by (2.11) D(t, y, x) > t−|y| and D(t, x, y) > t − |x| for x, y ∈ B n , and hence, in the notation of Theorem 2.13, It is also clear that D(t, y, x) ≥ t + |x| − |x − y|, and this implies that s 1 ≥ max{2(t + |x|) − |x − y|, 2(t + |y|) − |x − y|} = 2 max{t + |x|, t + |y|} − |x − y| and hence the inequality (2.19) follows. In the case of (2.20) we have D(t, y, x) > t + |x| and see that, in the notation of Corollary 2.16, s > 2(t + |x|) and (2.20) holds.

Remark.
(1) In several of the above results we have supposed that x, y ∈ B n \ {0} . If one of the points x, y were equal to 0 , then we would have a better result from the Schwarz lemma estimate (4.7).
(2) We see that the function h has a local minimum at t 1 = (3α) α λ α−1 n (β − α) −α . If t 1 ≥ 1 , then the inequality (2.19) yields the desired conclusion. The upper bound for T (n, K) follows by substituting the argument t 1 in the expression of h .
3.6. Graphical and numerical comparision of various bounds. The above bounds involve the Grötzsch ring constant λ n , which is known only for n = 2, λ 2 = 4. Therefore only for n = 2 we can compute the values of the bounds. Solving numerically the equation 4 · 16 1−1/K = h(t 1 ) for K we obtain K = 1.3089 . We give numerical and graphical comparison of the various bounds for the Mori constant. Tabulation of the various upper bounds for Mori's constant when n = 2 and λ 2 = 4 as a function of K: (a) Mori's conjectured bound 16 1−1/K , (b) the Anderson-Vamanamurthy bound 4 · 16 1−1/K , (c) the bound from (1.8). For K ∈ (1, 1.3089) the upper bound in (1.8) is better than the Anderson-Vamanamurthy bound and for K > 1.5946 the upper bound in (1.8) is better than the bound of Fehlmann and Vuorinen. Numerical values of the [FV] bound given in the table were computed with the help of the algorithm for ϕ K,2 (r) attached with [AVV1,p. 92,439].
K log(16 1−1/K ) log(4 · 16 1−1/K ) log(F V ) log(h(t 1 )) 1. For graphing and tabulation purposes we use the logarithmic scale. Note that the upper bound for M(2, K) given in [FV,Theorem 2.29] also has the desirable property that it converges to 1 when K → 1 , see Figure 2 the Hölder coefficient of f . Clearly HQ(f ) ≤ M(n, K). Theorem 2.13 yields, after dividing the both sides of the inequality in 2.13 by |x − y| α , the upper bound HQ(f ) ≤ HQ(K) for the Hölder quotient with (3.8) HQ(K) = sup{inf{U(t, x, y) : t ≥ 1} : x, y ∈ B n } , For n = 2 we compare HQ(K) to several other bounds (a) Mori's conjectured bound, (b) the FV bound, (c) the AV bound and give the results as a table and Figure 3. Because the supremum and infimum in (3.8) cannot be explicitly found we use numerical methods that come with Mathematica software. For the numerical tests we used for the supremum a sample of 100, 000 random points of the unit disk.
For K > 1.5946 the upper bound in (1.8) is better than the Fehlmann-Vuorinen bound.
the bound of the Mori conjecture. Note that the bound (3.8), based on a simulation with 100, 000 random points, gives the best estimate in the cases considered in the picture.

4.4.
Theorem. [Vu1,11.2] Let f : B n → R n be a nonconstant K-quasiregular mapping with f B n ⊂ B n and let α = K 1/(1−n) . Then for all x, y ∈ B n , where λ n is the same constant as in (1.5). If f (0) = 0 , then In the case of quasiconformal mappings with n = 2 formulas (4.5) and (4.7) also occur in [LV,p. 65] and formula (4.6) was rediscovered in [EMM,Theorem 5.1]. Comparing Theorem 4.4 to Theorem 1.10 we see that for n = 2 the expression K(ρ(x, y) + log 4) may be replaced with c(K) max{ρ(x, y), ρ(x, y) 1/K } , which tends to 0 when x → y and to ρ(x, y) when K → 1 , as expected.