Triangular Ratio Metric Under Quasiconformal Mappings in Sector Domains

The hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.


Introduction
Geometric function theory studies families of functions such as conformal maps, analytic functions as well as quasiconformal and quasiregular mappings defined in subdomains G of R n , n ≥ 2 .In this research, a key notion is an intrinsic distance, which is a distance between two points in the domain, specific to the domain itself and, in particular, its boundary [GH, HKV].In the planar case n = 2, such a distance is the hyperbolic distance that can be readily defined by use of a conformal mapping for a simply connected domain, but this does not generalize to higher dimensions.It is natural therefore to look for various extensions and generalizations of hyperbolic metrics.Twelve metrics recurrent in geometric function theory are listed by A. Papadopoulos in [P, pp. 42-48].
Many people have studied generalizations of hyperbolic metrics to subdomains of R n , n ≥ 3, and found hyperbolic type metrics, which share some but not all properties of the hyperbolic metric.In their study of quasidisks, F.W. Gehring and K. Hag [GH] apply the hyperbolic, quasihyperbolic, distance ratio, and Apollonian metrics.Very recently, the geometry of the quasihyperbolic metric has been studied by D. Herron and P. Julian [HJ], A. Rasila et al. [RTZ], S. Buckley and D. Herron [BH].Another hyperbolic type metric is the triangular ratio metric introduced by P. Hästö [H] and most recently studied by M. Fujimura et al. [FMV].The interrelations between these metrics have been investigated by P. Hästö, Z. Ibragimov, D. Minda, S. Ponnusamy and S. Sahoo [HIMPS].See also D. Herron et al. [HIM], and A. Aksoy et al. [AIW].
Our work is motivated by the recent progress of the study of intrinsic geometry of domains, of which the above papers and the monographs [GH, HKV, P] are examples.First, in Section 3, we find new inequalities between three different hyperbolic type metrics in sector domains of the complex plane and establish sharp forms of some earlier results in [CHKV, HVZ].In Section 4, we apply a rotation method involving Möbius transformations to obtain a sharp inequality between the triangular ratio metric and the hyperbolic metric in a sector with a fixed angle.Finally, in Section 5, we present our main result that provides a sharp distortion theorem for the triangular ratio metric under quasiconformal maps between two planar sector domains.

Preliminary facts
Define the following hyperbolic type metrics: The triangular ratio metric s and the point pair function These functions were studied in [CHKV, HVZ].Here, the domain G is a non-empty, open, proper and connected subset of R n and the notation d G (x) means the Euclidean distance dist(x, ∂G) = inf{|x − z| | z ∈ ∂G} between the point x and the boundary of G.Note that the point pair function is not always a metric [CHKV,Rmk 3.1 p. 689].
In this paper, we especially focus on the case where the domain G is an open sector S θ = {x ∈ C | 0 < arg(x) < θ} with an angle θ ∈ (0, 2π).In the limiting case θ = 0, we consider the strip domain With the notations presented above, we can also write the formulas for the hyperbolic metric in the upper half-plane and in the Poincaré unit disk, respectively [BM,(2.8) p. 15].Both of these metrics are invariant under a Möbius transformation h : In the two-dimensional plane, the definitions of hyperbolic metric can be simplified to where y is the complex conjugate of y.Moreover, in a planar simply-connected domain, the hyperbolic metric can be defined in terms of a conformal mapping of the domain onto the unit disk because the hyperbolic metric is invariant under conformal mappings [BM,Thm 6.3 p. 26].
The following inequalities between hyperbolic type metrics are already known: Theorem 2.5.For all θ ∈ (0, 2π) and x, y ∈ S θ , there is an analytical solution to the value of s S θ (x, y).
Proof.Consider a line l ⊂ R n , a half-line l 0 ⊂ l and two points x, y ∈ R n .Let x ′ be the point x reflected over the line l.
where l 1 , l 2 are the half-lines forming the sector S θ .

Hyperbolic Type Metrics
In this section, our main result is Theorem 3.13.This theorem provides sharp inequalities between the hyperbolic type metrics in a sector domain.First, we will show that certain equalities are possible.
. By [HKV,(1), p. 205], Remark 3.3.The metrics s G and j * G , and the point pair function p G are all invariant under the stretching z → rz by a factor r > 0, if the domain G is, for instance, H 2 , S θ or R n \{0}.
Proof.Consider the quotient We clearly need to find the maximal value of this quotient for each θ ∈ (π, 2π).First, fix the point z so that it gives the infimum inf z∈∂S θ (|x Now, we need to minimize the value of inf z∈∂S θ (|x−z|+|z −y|) compared to the numerator |x − y| + r of the quotient (3.5).This happens when the point z in the infimum is fixed to the origin, and x and y are on the different sides of the bisector of the sector Let us yet consider the distance |x − y| compared to the infimum in the quotient (3.5).

Suppose inf z∈∂S
, where l = d([x, y], {0}) so we see that, for a fixed sum |x| + |y|, the value of |x − y| is at maximum when |x| = |y|.
Because of these observations and by Remark 3.3, we can fix x = e hi and y = e (θ−h)i with some
Theorem 3.6.For a domain G R n , the sharp inequality holds for all x, y ∈ G.
Proof.According to [HKV,11.16(1),p. 203], the inequality p G (x, y)/ √ 2 ≤ s G (x, y) holds for all x, y ∈ G and this is sharp for p If we fix z ∈ ∂G so that it gives the infimum inf z∈∂G (|x It can be shown by differentiation that the quotient above attains its maximum value √ 2, Here, the equality holds for Remark Consequently, if s G (x, y) ≤ p G (x, y) holds for all x, y ∈ G, then G must be convex.
Theorem 3.9.For a fixed angle θ ∈ (0, π) and for all x, y ∈ S θ , the sharp inequality Proof.Consider the quotient Suppose without loss of generality that x = e hi and y = re ki with 0 < h ≤ k < θ and r > 0. Assume also that h < θ/2 < k, since otherwise p S θ (x, y) = s S θ (x, y).Now, the infimum in the quotient (3.10) is min{|x − y|, |x − y ′ |}, where x is the complex conjugate of x and y ′ is the point y reflected over the other side of the sector.Clearly, |x − y| = |1 − re (k+h)i | and |x − y ′ | = |1 − re (2θ−k−h)i |, so, to ensure the quotient is at minimum, we need to fix It follows that the quotient (3.10) is now Regardless of the exact values of r or θ, it can be shown with differentiation that the expression 1 + r 2 − 2r cos(θ − 2h) + 4r sin 2 (h) attains its minimum value when h = θ/4.Thus, the above quotient is minorized by .
By using differentiation again, we can show that the expression above attains its minimum value with r = 1.Thus, the minimum value of the quotient (3.10) is 2 cos(θ/4) .
Proof.Just like the proof of Theorem 3.9, fix x = e hi and y = re ki with 0 < h < θ/ 2 < k < θ and r > 0. Suppose also that k − h < π, for otherwise is the point y reflected over the other side of the sector.Thus, by symmetry, we can set Theorem 3.12.For a fixed angle θ ∈ (π, 2π) and for all x, y ∈ S θ , the sharp inequality s S θ (x, y) ≤ √ 2 sin(θ/4)p S θ (x, y) holds.
Proof.We are now interested in the maximum value of the quotient (3.10) for θ ∈ (π, 2π).
Theorem 3.14.The following inequalities hold for all x, y ∈ S 0 : y).Furthermore, in each case the constants are sharp.

Hyperbolic Metric in a Sector
The main result of this section is Corollary 4.9 which compares the triangular ratio metric and the hyperbolic metric of a sector domain.To prove it, we construct a conformal self-map of the sector, mapping two points in a general position to a pair of points, symmetric with respect to the bisector of the sector angle.Because conformal maps preserve the hyperbolic distance, under this mapping the hyperbolic distance remains invariant whereas the triangular ratio distance may change.This enables us to reduce the comparison of these two metrics to the case when the points are symmetric with respect to the bisector.Proposition 4.1.Let x, y ∈ H 2 be two distinct points, let L(x, y) be the line through them, and let the angle of intersection between L(x, y) and the real axis be α and suppose that α ∈ (0, π/2).Then there are two circles S 1 (c 1 , r 1 ) and S 1 (c 2 , r 2 ), centered at the real axis and orthogonal to each other, such that x, y ∈ S 1 (c 1 , r 1 ) and c 2 = L(x, y) ∩ R.
Proof.First, note that there is a conformal mapping h : S 0 → H 2 , h(z) = e z .By using the Möbius transformation g of Lemma 4.2, we can create a conformal mapping for some k ∈ (0, 1).Just like in the proof of Lemma 4.5, it follows that Consider now the quotient The quotient (4.13) is therefore sin(kπ/2)/k.By differentiation, it can be shown that this result is decreasing with regards to k, so its minimum value is lim k→1 − (sin(kπ/2)/k) = 1 and its maximum value lim k→0 + (sin(kπ/2)/k) = π/2.
Remark 4.15.The inequalities of Theorem 4.12 and Corollary 4.14 are the same as the inequalities of Corollaries 4.9, 4.10 and 4.11 when θ → 0 + .

s-Metric in Quasiconformal Mappings
The main result of this section and the whole paper is Corollary 5.7.First, we introduce a general result related to the triangular ratio metric under quasiconformal mappings and then we develop it further with the inequalities of Corollary 4.9.At the end of this section, we also consider the triangular ratio metric under a conformal mapping between sectors.The behaviour of the triangular ratio metric under Möbius transformations and quasiconformal mappings has been studied earlier; see [CHKV,Thms 1.2 & 1.3 p. 684;Cor. 3.30 & Thm 3.31 p. 697], [HVZ, Thm 4.7 p. 1144;Thm 4.9 p. 1146].
For the definition and basic properties of K-quasiconformal homeomorphisms, the reader is referred to [V,Ch.2].We start with two preliminary results.
The main results of this section are based on the following recent form of the quasiconformal Schwarz lemma: Theorem 5.4.Let G 1 and G 2 be simply-connected domains in R 2 and f : for all x, y ∈ G 1 where c(K) is as in [HKV,Thm 16.39,p. 313], [WV,Theorem 3.6].
Corollary 5.6.Let G 1 and G 2 be simply-connected domains in R 2 and f : and, applying Theorem 5.3 with C = c(K), we will have Corollary 5.7.If α, β ∈ (0, 2π) and f : S α → S β = f (S α ) is a K-quasiconformal homeomorphism, the following inequalities hold for all x, y ∈ S α . (1) Proof.Follows from Corollaries 4.9 and 5.6, and the fact that the inverse mapping f −1 of a K-quasiconformal mapping f is another K-quasiconformal mapping.
Remark 5.9.Note that the inequality in Corollary 5.8 reduces to an identity if K = 1.
By differentiation, it can be proved that this quotient is monotonic with respect to k and its extreme values are It only depends on whether α ≤ β or not, which one of these extreme values is the minimum and which one the maximum, so our theorem follows.
and the unit ball B n = B n (0, 1).Here, B n (x, r) is the open ball with the Euclidean metric, B n (x, r) is the corresponding closed ball and S n−1 (x, r) is the boundary ∂B n (x, r).For two distinct points x, y, L(x, y) is the Euclidean line passing through them.