Möbius metric in sector domains

The Möbius metric δG is studied in the cases, where its domain G is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Möbius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Möbius metric and its connection to the hyperbolic metric in polygon domains.


Introduction
One of the most important concepts in the geometric function theory is the intrinsic distance.It means that, given two points in a domain, we do not only consider how close these points are to each other but also how they are located with respect to the boundary of the domain.In order to measure these kinds of distances, we need to use suitable intrinsic or hyperbolic type metrics, which have been recently studied, for instance, in [1,5,6,7,9,12,13,14].
In this article, we focus on one of these intrinsic metrics, which is defined as follows: For any domain G ⊂ R n = R n ∪ {∞} whose complement (R n \G) contains at least two points, let the Möbius metric be the function where |a, x, b, y| is the cross-ratio defined in (2.1).This metric was first introduced in [17, pp. 115-116] and then later studied more extensively by P. Seittenranta in his PhD thesis [15,Def. 1.1,p. 511], which is why it is sometimes also referred to as Seittenranta's metric.
Due to the Möbius invariance of the cross-ratio, the distances defined with the Möbius metric are preserved under Möbius transformations, which is one of the most useful properties of this metric.However, there are still numerous open questions concerning this metric.For instance, while it is known that the value of this metric is equal to that of the hyperbolic metric ρ or the distance ratio metric j in some special cases, see Theorems 2.3 and 3.1, the Möbius metric is studied very little in other kinds of domains.To fill this gap, our aim here is to find more information about the Möbius metric in the cases where the domain G is either an open sector of the complex plane or a polygon.
The structure of this article is as follows.First, in Section 3, we combine some already known inequalities to create some initial bounds for the Möbius metric in a general domain.Then, in Section 4, we study the Möbius metric defined in an open sector by showing how the supremum of the cross-ratio in its definition (1.1) can be found.These results will be used in Section 5, where we introduce bounds for the Möbius metric in terms of the hyperbolic metric in a sector and prove Theorem 1.2.In Section 6, we apply these results and prove bounds for the distortion of the Möbius metric under quasiregular mappings of the unit disk into sector domains.Finally, in Section 7, we utilise the recent computational methods from [11] to experimentally study the inequalities between the Möbius and hyperbolic metric in polygon domains and formulate a few conjectures.

Preliminaries
First, define the following notations for the Euclidean metric.Let the distance from a point x ∈ R n to a non-empty set Let the Euclidean diameter of a nonempty set F be d(F ) and the Euclidean distance between two non-empty separate sets F 0 , F 1 be d(F 0 , F 1 ).Furthermore, denote the Euclidean open ball with a center x ∈ R n and a radius r > 0 by B n (x, r), the corresponding closed ball by B n (x, r) and its boundary sphere by S n−1 (x, r).
Let R n = R n ∪ {∞} be as in the introduction, and also denote C n = C n ∪ {∞}.For all distinct points x, y ∈ R n , define the spherical (chordal) metric as in [5, (3.6), p. 29]: For any four distinct points a, b, c, d ∈ R n , define the cross-ratio as [5, (3.10), p. 33] Other than the Möbius metric, we will be needing a few other hyperbolic type metrics.Define the upper half-space and the open sector S θ = {x ∈ C\{0} | 0 < arg(x) < θ} with an angle θ ∈ (0, 2π).Here, arg(x) ∈ [0, 2π) denotes the principal branch of the argument of a complex number x ∈ C\{0}.Now, we can define the hyperbolic metric in these three domains by using the following formulas, respectively [5, (4.8), p. 52 & (4.14), p. 55]: From these formulas, it follows that: where y is the complex conjugate of y.
For a domain G R n , the distance ratio metric [17, p. 25] introduced by Gehring and Osgood [3] is the function j As noted in [7, 2.2, p. 1123 & Lemma 2.1, p. 1124], this metric can be used to define another metric, the so called Furthermore, the quasihyperbolic metric introduced by Gehring and Palka in [4] is defined as the function where Γ xy consists of all the rectifiable curves in G joining x and y.Consider yet the which was originally introduced by P. Hästö in 2002 [8].
The following result expresses the main property of the Möbius metric:

General inequalities
In this section, we will briefly review a few already existing inequalities and show how they can be used to create bounds for the Möbius metric.Note that the inequalities found here concern mostly the situation, in which the shape of the domain G is not known.For instance, Corollary 3.5 gives us an inequality for a simply connected uniform domain G, but its constants are probably not very sharp when compared to those that could be obtained when knowing the exact shape of the domain G.
for all x, y ∈ G.
Proof.Follows from Theorems 3.3 and 3.2, and Definition 3.4.
Now, let us find some bounds for the Möbius metric with the triangular ratio metric and the j * -metric in the cases of both a convex domain G and a non-convex one.
Proof.(1) Follows from Theorem 3.1 and the definition of j * -metric.
(2) Follows from the first inequality and Lemma 3.6.
Finally, let us consider the case where the domain G is an open sector.Note that neither the inequalities of Corollary 3.10 nor Corollary 3.12 have the best possible constants.In fact, they are only used to prove our main result, Theorem 1.2, in Section 5. Theorem 3.9.[14, Cor.4.9, p. 9] For a fixed angle θ ∈ (0, 2π) and for all x, y ∈ S θ , the following results hold: (1) Furthermore, these bounds are also sharp.Corollary 3.10.For all points x, y ∈ S θ , the following inequalities hold: Proof.Follows from Corollary 3.8 and Theorem 3.9.

Möbius metric in open sector
In this section, our aim is to find ways to estimate the value of the Möbius metric defined in an open sector S θ .To do this, we will study the supremum of the cross-ratio needed in the definition of the metric δ S θ , in both cases where the angle θ is less than π and greater than π.The main result of this section is Corollary 4.13 but, in order to prove it, we need to consider several other results first.
Proof.Since δ H 2 (x, y) = sup a,b∈R log(1 + |a, x, b, y|), both results can be verified by the fact that δ G = ρ G for G ∈ {B n , H n } according to Theorem 2.3.Lemma 4.2.For all points x, y in an open sector S θ with an angle 0 < θ < π such that arg(x) ≤ arg(y) and |x| = |y| = r > 0, there is a Möbius transformation f that maps S θ onto the lens-shaped domain Proof.Define the function f : By the general properties of Möbius transformations, f preserves the angles and must turn a line into a circle if any three points chosen from it are no longer collinear after the transformation.Thus, f maps two sides of the sector S θ onto two circular arcs that are symmetric with respect to the both coordinate axes, meet each other in the points f (0) and f (∞) at an angle of θ, and out of which one contains the point i and the other one the point −i.See Figure 1.
Consider a circle S 1 ((1 − u)i, u), u > 1.Clearly, it is symmetric with respect to the coordinate axes and i ∈ S 1 ((1 − u)i, u).Using simple trigonometry, it can be calculated that the two interior angles of the figure consisting of the real axis and the circular arc If the value of this angle is θ/2, we can solve that By combining all our observations made above, we will have that, for all 0 < θ < π, where u is as in (4.5).The final part of the lemma is very trivial: From the behaviour of the points in (4.4), we see that the transformation f maps the circle S 1 (0, r) onto the imaginary axis and, if x, y ∈ S 1 (0, r) such that 0 < arg(x) ≤ arg(y) < θ, then clearly The result of Lemma 4.2 is very useful because it follows from the Möbius invariance of the cross-ratio that the value of the Möbius metric between x, y ∈ S θ can be calculated in the lens-shaped symmetric domain f (S θ ) for f (x), f (y), see Figure 1.
Proof.Let f be the Möbius transformation defined in (4.3), under which the open sector S θ with an angle 0 < θ < π is mapped onto a lens-shaped domain f (S θ ).For all points x, y ∈ iR ∩ B 2 , choose a, b ∈ f (∂S θ ) so that the cross-ratio |a, x, b, y| is at greatest.Note that, for all points u ∈ iR ∩ B 2 and v ∈ C\B 2 , the inequality Because f preserves the cross-ratio as a Möbius transformation, we can now show that, for all x, y ∈ S θ such that |x| = |y| = r and arg(x) ≤ arg(y), Note that Theorem 4.6 does not hold in the case θ > π, as the following example shows.
However, we can still use Lemma 4.2 to calculate the supremum of the cross-ratio in the Möbius metric for points x, y in a sector S θ with θ > π, as can be seen from the following result.
Corollary 4.11.For any open sector S θ with an angle π < θ < 2π, there is a Möbius transformation f that maps S θ onto the domain Proof.Let the Möbius transformation f be as in (4.3) with, for instance, r = 1.The proof now goes just like that of Theorem 4.6, but it must be noted that f maps the sides of S θ onto circular arcs that meet each other at an angle θ > π.Thus, f (S θ ) must be an union of two disks B 2 ((1 − u)i, u) and B 2 ((u − 1)i, u) instead of their intersection and 1/2 < u < 1 now, see Figure 2. The final part of the proof follows from the Möbius invariance of the cross-ratio.
The results of this section about the supremum of the cross-ratio give us information about the values of the Möbius metric δ S θ defined in a sector domain.where the equality holds whenever θ ≤ π.
Proof.Let x = re ui and y = re vi with 0 < u ≤ v < θ.By Theorem 4.6 and Proposition 4.1(1), the supremum sup a,b∈∂S θ |a, x, y, b| is now found by choosing a = r and b = re θi if θ ≤ π, and these choices of a, b give a lower limit for the supremum if θ > π.The result follows now directly from the definition of δ S θ (x, y).

Möbius metric and hyperbolic metric in open sector
In this section, we will study the connection between the Möbius metric and the hyperbolic metric in an open sector S θ with an angle 0 < θ < 2π.The main result of this section is Corollary 5.11, which will be used to prove Theorem 1.2.However, in order to derive these results, we need to define the following quotient.
For all 0 < k < 1 and 0 < θ < 2π, define The quotient above is very much needed here because it equals to the value of the quotient between the Möbius metric and the hyperbolic metric in certain cases, as will be shown in the next lemma.
Proof.Let the mappings f, g, h be as in Lemma 5.3 and the proof of Lemma 5.4.Note that, even though the mapping f of does not necessarily preserve the distance δ S θ (x, y), by Theorem 2.3 and the conformal invariance of the hyperbolic metric, for all points x, y ∈ S θ .It follows from this that inf which leads to the result of our corollary by Lemmas 5.2 and 5.4.
Before studying the values of the quotient Q(k, θ), consider yet the following proposition.
It can be verified that the number 1 in the inequalities of Theorem 5.7 is the best possible constant by showing that it is the limit value of the quotient Q(k, θ) whenever k → 1 − .However, the bound (π sin(θ/2)/θ) 2 in Theorem 5.7 does not seem to be sharp.According to several numerical test, the quotient Q(k, θ) is monotonic with respect to k, either decreasing when 0 < θ ≤ π or increasing when π ≤ θ < 2π, which would lead into the following result.
Note that Corollary 5.11 does not offer an upper bound for the metric δ S θ in terms of ρ S θ in the case θ > π.As stated in Corollary 5.5, Q(k, θ) is only a lower limit for the quotient δ S θ (x, y)/ρ S θ (x, y), so the result of Theorem 5.7 gives us this upper bound.Specifically, even though Q(k, θ) ≤ 1 for π < θ < 2π, the inequality δ S θ (x, y) ≤ ρ S θ (x, y) does not hold, as will be shown next.
Proof.For x = e (1−k)θi/2 and y = e (1+k)θi/2 with 0 < k < 1, the distance ρ S θ is as in the proof of Lemma 5.2 and, consequently, Finally, we can combine the inequalities of Corollary 5.11 with our earlier results from Section 3 in order to show that Theorem 1.2 holds.5.14.Proof of Theorem 1.2.Follows directly from Corollaries 3.10, 3.12 and 5.11.Note that Theorem 1.2 only contains the best ones out of the bounds found and, for instance, the lower bound for δ S θ (x, y) in Corollary 3.10(1) is never better than the one in Corollary 5.11 (1).Similarly, it can be shown that Corollary 5.11 has always better lower bounds than Corollary 3.12.

Möbius metric under quasiregular mappings
In this section, we will yet briefly consider the behaviour of the Möbius metric under Kquasiregular mappings.This topic has already been researched in [15], see for instance [15,Thm 5.12, pp.528-529], but we can improve the existing results with our new bounds for the Möbius metric in sector domains.However, let us first define all the concepts needed.Definition 6.1.[5, pp. 289-288] Choose a domain G ⊂ R n and let the function f : G → R n be ACL n , as defined in [5, p. 150].Suppose that there exists a constant K ≥ 1 such that the inequality where J f (x) is the Jacobian determinant of f at point x ∈ G, holds a.e. in G. Then the function f is called quasiregular and the smallest constant K ≥ 1 fulfilling the inequality (6.2) is the outer dilatation of f .Similarly, the inner dilatation of f is the smallest constant K ≥ 1 such that the inequality where K I (f ) and K O (f ) are the inner and the outer dilatation of f , respectively.
The Schwarz lemma is one of the most well-known results in the distortion theory and, while its original version is about the distortion of the Euclidean metric under holomorphic functions, there exists the following modified version of the Schwarz lemma that tells about the distortion of the hyperbolic metric under K-quasiregular mappings.
holds for all x, y ∈ G. Furthermore, in the two-dimensional case n = 2, ( as in [5,Thm 16.39,p. 313].Here, c(K) → 1 when K → 1 and, by the conformal invariance of the hyperbolic metric, the result (3) also holds for any two simply connected planar domains G, G because they can be mapped conformally onto the unit disk B 2 .
A similar result holds for a non-convex sector.
Remark 6.8.Neither Corollary 6.6 nor Corollary 6.7 offers a bound for the distortion that is always better than the result of the other corollary, which can be seen by studying the case where θ → π − and c(K) = K = 1 for varying points x, y ∈ S θ .

Möbius metric in polygon
In this section, we will introduce a few open questions related to the Möbius metric inside a polygon domain.Especially, we are interested in the inequality between the Möbius metric and the hyperbolic metric defined in a polygon.All the computational findings and Figure 3 have been made with MATLAB programs from [11].
Even though the inequality ρ S θ (x, y) ≤ δ S θ (x, y) holds in all convex sectors by Theorem 1.2 and these metrics are equivalent in such convex domains as the unit disk and the upper half-plane, our computer experiments verify that neither of metrics is always greater than or equal to the other in all polygonal domains, not even in all convex polygons.Conjecture 7.1.If G R 2 is any bounded polygonal domain, there are always some points x, y, u, v ∈ G such that ρ G (x, y) < δ G (x, y) and ρ G (u, v) > δ G (u, v).
However, the values of these two metrics do not differ very much from each other in the domain G k shaped like a regular convex k-gon with k vertices e p2πi/k , p = 0, 1, ..., k − 1, especially as the value of k grows and this domain resembles more and more the unit disk, where these metrics are equivalent.), when G is the polygon with vertices 1, e 0.95πi/3 , 0.1e πi/3 , e 1.05πi/3 , e 2πi/3 , e πi , e 4πi/3 , e 5πi/3 , the point x is fixed as 0, and y varies inside G.
Furthermore, recall from Corollary 3.5 that the inequality δ G (x, y) ≤ 4ρ G (x, y) holds for all points x, y in a simply-connected domain G, and our computer tests suggests that this result can be improved in the case of polygonal domains.for all points x, y in the domain x, y, so computing the maximum value of the quotient ρ G (x, y)/(2δ G (x, y) gives an lower bound for the uniformity constant A G .
7) holds.It follows from this that |x − a| ≤ |a − b| holds, because otherwise replacing a either by i or −i would give a greater value for the cross-ratio |a, x, b, y|.

Figure 3
Figure 3 contains an example of a non-convex polygon where the values of the quotient δ G (x, y)/ρ G (x, y) vary at least on the interval [0.73, 1.64].Note that, based on our computer tests, the latter constant in the inequality of Conjecture 7.3 can be replaced with a smaller one when considering only convex domains.Another interesting notion is that by Corollary 3.5 the uniformity constant A G of a domain G is fulfills A G ≥ ρ G (x, y)/(2δ G (x, y) 2 is decreasing with respect to t, by [5, Thm B.2, p. 465], f 4 (t)/f 5 (t) is decreasing, too.