aa r X i v : . [ m a t h . M G ] N ov INTRINSIC QUASI-METRICS
OONA RAINIO
Abstract.
The point pair function p G defined in a domain G ( R n is proven to bea quasi-metric with the best possible constant √ /
2. For a convex domain G ( R n , anew intrinsic quasi-metric called the function w G is introduced. Several sharp resultsare established for these two quasi-metrics, and their connection to the triangular ratiometric is studied. Introduction
In geometric function theory, one of the key concepts is an intrinsic distance. Thisnotion means a distance between two points fixed in a domain that not only dependson how close these points are to each other but also takes into account how they arelocated with respect to the boundary of the domain. A well-known example of an intrinsicmetric is the hyperbolic metric [1], which is conformally invariant and therefore has severalproperties useful in the study of the distortion under different functions.However, the hyperbolic metric is not the only metric that can be used to model theintrinsic distances and not even its key features are unique. On the contrary, it has severalgeneralizations that share its main properties and whose behaviour and qualities have alsobeen widely studied [5, Ch. 5, pp. 67-81]. Especially during the past thirty years, varioushyperbolic type metrics have been introduced, see [2, 5, 7, 9, 10, 12].This often raises the question about the reason for introducing new metrics and studyingthem instead of just focusing on those already existing. To answer this, it should be firstnoted that the slightly different definitions of the intrinsic metrics mean that they haveunique advantages and suit for diverse purposes. Consequently, new metrics can be usedto discover various intricate features of geometric entities that would not be detectedwith some other metrics. For instance, many hyperbolic type metrics behave slightlydifferently under quasiconformal mappings and analysing these differences can give us abetter understanding of how such mappings really work [11].Furthermore, new metrics can also bring information about the already existing metrics.Calculating the exact value of the hyperbolic metric in a domain that cannot be mappedonto the unit disk with a conformal mapping is often impossible but we can estimate it byusing other intrinsic metrics with simpler definitions [5, Ch. 4.3, pp. 59-66]. However, in
File: main.tex, printed: 2020-11-5, 1.49
Mathematics Subject Classification.
Primary 51M10; Secondary 51M16.
Key words and phrases.
Hyperbolic geometry, intrinsic geometry, intrinsic metrics, quasi-metrics, tri-angular ratio metric. order to do this, we need to know the connection between the different metrics consideredand to be able to create upper and lower bounds for them. Finding sharp inequalities forintrinsic metrics can often help us with some related applications and, for instance, in theestimation of condenser capacities [5, Ch. 9, pp. 149-172].Another noteworthy motivation for studying several different metrics is that their in-equalities can tell us more about the domain where the metrics are defined. The definitionfor a uniform domain can be expressed with an inequality between the quasihyperbolicmetric and the distance ratio metric as in [5, Def. 6.1, p. 84]. Similarly, some otherinequalities can be used to determine whether the domain is, for instance, convex or not,like in Theorem 3.17. Further, Corollary 3.20 even shows an equality between metricsthat serves as a condition for when the domain can be mapped onto a half-space withjust Euclidean isometries.In this paper, we consider two different intrinsic quasi-metrics. By a quasi-metric , wemean a function that fulfills all the conditions of a metric otherwise but only a relaxedversion of the triangle inequality instead of the inequality itself holds for this function, seeDefinition 2.1 and the inequality (2.2). The first quasi-metric considered is the point pairfunction introduced by Chen et al. in 2015 [2], and the other quasi-metric is a functiondefined for the first time in Definition 4.1 in this paper. We also study the triangularratio metric introduced by P. H¨ast¨o in 2002 [7] for one of the main results of this paperis showing how our new quasi-metric can be used to create a very good lower bound forthis metric, especially in the case where the domain is the unit disk.The structure of this paper is as follows. In Section 3, we prove the exact constant withwhich the point pair function is a quasi-metric and show how it can be used together withthe triangular ratio metric to tell us about the shape of the domain. Then, in Section4, we introduce a new quasi-metric and also inspect its connection to triangular ratiometric. Finally, in Section 5, we focus on the case where the domain is the unit disk andfind several sharp inequalities between different hyperbolic type metric and quasi-metricsconsidered. Especially, we investigate how the new quasi-metric can be used to estimatethe value of the triangular ratio metric in the unit disk, see Theorem 5.7 and Conjecture5.7.
Acknowledgements.
This research continues my work with Professor Matti Vuorinenin [12, 13, 14]. I am indebted to him for all guidance and other support. My research wasalso supported by Finnish Concordia Fund.2.
Preliminaries
In this section, we will introduce the necessary definitions and some basic results relatedto them but let us first recall the definition of a metric.
Definition 2.1.
For any non-empty space G , a metric is a function d : G × G → [0 , ∞ )that fulfills the following three conditions for all x, y, z ∈ G :(1) Positivity: d ( x, y ) ≥
0, and d ( x, y ) = 0 if and only if x = y , NTRINSIC QUASI-METRICS 3 (2) Symmetry: d ( x, y ) = d ( y, x ),(3) Triangle inequality: d ( x, y ) ≤ d ( x, z ) + d ( z, y ) . A quasi-metric is a function d that fulfills the definition above otherwise, but insteadof the triangle inequality itself, it only fulfills the inequality d ( x, y ) ≤ c ( d ( x, z ) + d ( z, y ))(2.2)with some constant c > G ( R n is some domain. Forall x ∈ G , the Euclidean distance d ( x, ∂G ) = inf {| x − z | | z ∈ ∂G } will be denoted by d G ( x ). The Euclidean balls and spheres are written as B n ( x, r ) = { y ∈ R n | | x − y | < r } , B n ( x, r ) = { y ∈ R n | | x − y | ≤ r } and S n − ( x, r ) = { y ∈ R n | | x − y | = r } . If x or r isnot specified otherwise, suppose that x = 0 and r = 1. Furthermore, the Euclidean linepassing through points x, y ∈ R n is denoted by L ( x, y ) and the unit vectors by { e , ..., e n } .In this paper, we focus on the cases where the domain G is either the upper half-plane H n = { ( x , ..., x n ) ∈ R n | x n > } , the unit ball B n = B n (0 ,
1) or the open sector S θ = { x ∈ C | < arg( x ) < θ } with an angle θ ∈ (0 , π ). The hyperbolic metric can bedefined in these cases with the formulasch ρ H n ( x, y ) = 1 + | x − y | d H n ( x ) d H n ( y ) , x, y ∈ H n , sh ρ B n ( x, y )2 = | x − y | (1 − | x | )(1 − | y | ) , x, y ∈ B n ,ρ S θ ( x, y ) = ρ H ( x π/θ , y π/θ ) , x, y ∈ S θ , see [5, (4.8), p. 52 & (4.14), p. 55]. In the two-dimensional unit disk, we can simply writeth ρ B ( x, y )2 = (cid:12)(cid:12)(cid:12)(cid:12) x − y − xy (cid:12)(cid:12)(cid:12)(cid:12) , where y is the complex conjugate of y .For any domain G ( R n , define the following hyperbolic type metrics and quasi-metric:(1) [2, (1.1), p. 683] The triangular ratio metric s G : G × G → [0 , ,s G ( x, y ) = | x − y | inf z ∈ ∂G ( | x − z | + | z − y | ) , (2) [6, 2.2, p. 1123 & Lemma 2.1, p. 1124] the j ∗ G -metric j ∗ G : G × G → [0 , ,j ∗ G ( x, y ) = | x − y || x − y | + 2 min { d G ( x ) , d G ( y ) } , (3) [2, p. 685], [6, 2.4, p. 1124] the point pair function p G : G × G → [0 , ,p G ( x, y ) = | x − y | p | x − y | + 4 d G ( x ) d G ( y ) . O. RAINIO
Remark 2.3.
All three functions listed above are invariant under all similarity maps.In particular, when defined in a sector S θ , they are invariant under a reflection over thebisector of the sector and a stretching x r · x with any r >
0. Consequently, this allowsus to make certain assumptions when choosing the points x, y ∈ S θ .The metrics introduced above fulfill the following inequalities. Lemma 2.4. [6, Lemma 2.1, p. 1124; Lemma 2.2 & Lemma 2.3, p. 1125 & Thm 2.9(1),p. 1129]
For any subdomain G ( R n and all x, y ∈ G , the following inequalities hold: (1) j ∗ G ( x, y ) ≤ p G ( x, y ) ≤ √ j ∗ G ( x, y ) , (2) j ∗ G ( x, y ) ≤ s G ( x, y ) ≤ j ∗ G ( x, y ) .Furthermore, if G is convex, then for all x, y ∈ G (3) s G ( x, y ) ≤ √ j ∗ G ( x, y ) . Lemma 2.5. [5, p. 460]
For all x, y ∈ G ∈ { H n , B n } , (1) th ρ H n ( x, y )4 ≤ j ∗ H n ( x, y ) ≤ s H n ( x, y ) = p H n ( x, y ) = th ρ H n ( x, y )2 ≤ ρ H n ( x, y )4 , (2) th ρ B n ( x, y )4 ≤ j ∗ B n ( x, y ) ≤ s B n ( x, y ) ≤ p B n ( x, y ) ≤ th ρ B n ( x, y )2 ≤ ρ B n ( x, y )4 . Point pair function
In this section, we will focus on the point pair function. The expression for this functionwas first introduced in [2, p. 685], but it was named and researched further in [6]. Itwas noted already in [2, Rmk 3.1 p. 689] that the point pair function defined in the unitdisk is not a metric because it does not fulfill the triangle inequality for all points of thisdomain. However, the point pair function offers a good upper bound for the triangularratio metric in convex domains [5, Lemma 11.6(1), p. 197] and, by Lemma 2.5, it alsoserves as a lower bound for the expression th( ρ G ( x, y ) /
2) if G ∈ { H n , B n } so studying itsproperties more carefully is relevant.It is very easy to show that there is a constant c > Corollary 3.1.
For all domains G ( R n , the point pair function p G is a quasi-metricwith a constant less than or equal to √ .Proof. It follows from Lemma 2.4(1) and the fact that the j ∗ G -metric is always a metricthat p G ( x, y ) ≤ √ j ∗ G ( x, y ) ≤ √ j ∗ G ( x, z ) + j ∗ G ( z, y )) ≤ √ p G ( x, z ) + p G ( z, y )) . (cid:3) However, even for an arbitrary domain G ( R n , the constant √ √ /
2. Now, let us introduce two quite simple results in order to prove Theorem3.7.
NTRINSIC QUASI-METRICS 5
Proposition 3.2.
If the points x, z ⊂ R n are fixed and x ′ ∈ S n − ( x, r ) for some constant r > , then | x − z | p | x − z | + 4 | x − x ′ || z − x ′ | ≥ | x − z || x − z | + 2 | x − x ′ | and the equality here holds if and only if | z − x ′ | = r + | x − z | .Proof. Since the distances | x − z | and | x − x ′ | = r are fixed, the minimum value of the firstquotient above is obtained by maximising the distance | z − x ′ | . By the triangle inequality, | z − x ′ | ≤ | x − z | + | x − x ′ | and the equality here holds if and only if x ′ ∈ L ( x, z ) suchthat x ∈ [ z, x ′ ]. Thus, the result follows. (cid:3) Lemma 3.3.
Out of all possible domains G ( R n , the value of the supremum sup x,y,z ∈ G p G ( x, y ) p G ( x, z ) + p G ( z, y )(3.4) is at greatest if G = [ x ′ , y ′ ] ( R and, in this case, the supremum is obtained when x ′ < x ≤ z ≤ y < y ′ .Proof. Let us first consider the question without specifying the domain G ( R n or itsboundary in any way. Instead, fix distinct points x, y ∈ R n and choose distances d , d > | x − y | p | x − z | + 4 d d (3.5)is fixed.Next, choose z ∈ R n and suppose that there are some points x ′ ∈ S n − ( x, d ) and y ′ ∈ S n − ( y, d ) but do not fix these points x ′ , y ′ otherwise. Just like in Proposition 3.2,the minimum value of the sum | x − z | p | x − z | + 4 d min {| z − x ′ | , | z − y ′ |} + | z − y | p | z − y | + 4 d min {| z − x ′ | , | z − y ′ |} . (3.6)is obtained by maximizing the distances | z − x ′ | ≤ d + | x − z | and | z − y ′ | ≤ d + | z − y | .By symmetry, let us assume that d + | x − z | ≤ d + | z − y | . The sum (3.6) becomes now | x − z || x − z | + 2 d + | z − y | p | z − y | + 4 d ( d + | x − z | ) . Since this sum is increasing with respect to | x − z | and | z − y | , we need to choose z ∈ [ x, y ]to minimize it and, thus, the line L ( x, y ) must contain the points x ′ , x, z, y, y ′ in this order.Finally, consider some domain G ( R n . For all x, y ∈ G , there are some points x ′ , y ′ ∈ ∂G such that d G ( x ) = d = | x − x ′ | and d G ( y ) = d = | y − y ′ | . The expressionfor p G ( x, y ) is equivalent to the quotient (3.5), and we need to find the minimum valuefor the sum p G ( x, z ) + p G ( z, y ). From the expression of the function p G , we see thatthis sum is is clearly decreasing with respect to d G ( z ). There only needs to be points O. RAINIO x ′ , y ′ ∈ ∂G defining the distances d and d , so let d G ( z ) = min {| z − x ′ | , | z − y ′ |} . Thesum p G ( x, z ) + p G ( z, y ) is now like in (3.6), and it follows from above that it is minimizedwhen x ′ , x, z, y, y ′ ∈ L ( x, y ) in this order. Furthermore, we can now set n = 1 withoutloss of generality and the result follows. (cid:3) With the lemma above, we can now prove the best constant with which the point pairfunction is a quasi-metric, assuming that the domain is not fixed.
Theorem 3.7.
For all domains G ( R n , the point pair function p G is a quasi-metric witha constant at most √ / and this constant here is the best possible.Proof. We are interested in the greatest value of the quotient (3.4) so, by Lemma 3.3, wecan suppose that G = [ x ′ , y ′ ] ( R and x ′ < x ≤ z ≤ y < y ′ . Without loss of generality,fix | x − y | = 1 and 0 < d G ( x ) ≤ d G ( y ). We can now choose | x − z | = k and | z − y | = 1 − k for some 0 ≤ k ≤
1. Because d G ( y ) ≤ d G ( x ) by the triangle inequality, we can alsowrite d G ( y ) = d G ( x ) + h for some 0 ≤ h ≤
1. Furthermore, it clearly now holds that d G ( z ) = min { k + d G ( x ) , − k + d G ( x ) + h } . (3.8)Because k + d G ( x ) ≤ − k + d G ( x ) + h ⇔ k ≤ h , we will have by the equality (3.8) that d G ( z ) = ( k + d G ( x ) , if k ≤ (1 + h ) / − k + d G ( x ) + h, if k > (1 + h ) / p G ( x, z ) + p G ( x, y ) can be described with the function l ( k, h ) = kk + 2 d G ( x ) + 1 − k p (1 − k ) + 4( d G ( x ) + h )( k + d G ( x )) , if k ≤ h ,k p k + 4 d G ( x )(1 − k + d G ( x ) + h ) + 1 − k − k + 2( d G ( x + h )) , if k > h . Denote now l ( k, h ) = kk + 2 d G ( x ) + 1 − k p (1 − k ) + 4( d G ( x ) + h )( k + d G ( x )) ,l ( k, h ) = k p k + 4 d G ( x )(1 − k + d G ( x ) + h ) + 1 − k − k + 2( d G ( x ) + h ) ,j ( h ) = 1 p d G ( x )( d G ( x ) + h ) . Here, the function j ( h ) is the value of the distance p G ( x, y ), where d G ( y ) = d G ( x ) + h . NTRINSIC QUASI-METRICS 7
We will have j ( h ) l ( k, h ) = √ d G ( x )( d G ( x )+ h ) kk +2 d G ( x ) + − k √ (1 − k ) +4( d G ( x )+ h )( k + d G ( x )) , (3.9) j ( h ) l ( k, h ) = √ d G ( x )( d G ( x )+ h ) k √ k +4 d G ( x )(1 − k + d G ( x )+ h ) + − k − k +2( d G ( x )+ h ) . (3.10)Consider the quotient (3.9). Increasing the value of h lessens quite similarly the numeratorand the second term in the denominator. However, the first term of the denominator isclearly larger than the second term and it does not depend on h . Thus, we can see thatthe quotient (3.9) is decreasing with respect to h . With a similar argument, we can alsoshow that the quotient (3.10) is increasing with respect to h .From these observations, it follows that(3.11) sup ≤ h ≤ , ≤ k ≤ j ( h ) l ( k, h )= max ( sup ≤ h ≤ , ≤ k ≤ (1+ h ) / j ( h ) l ( k, h ) , sup ≤ h ≤ , (1+ h ) / 1) = l ( k, k − 1) for all1 / ≤ k ≤ 1. This is because h = 2 k − k = ( h + 1) / l ( k, h ) is continuous here. Next, we need to calculate the supremums above and find outwhich one of them is greater.Let us first find the minimum value of the function l ( k, 0) for 0 ≤ k ≤ / 2. From thedefinition of l ( k, h ), we will have l ( k, 0) = kk + 2 d G ( x ) + 1 − k p (1 − k ) + 4 d G ( x )( k + d G ( x )) . O. RAINIO By differentiation, ∂∂k l ( k, 0) = 2 d G ( x )( k + 2 d G ( x )) − d G ( x )( k + 2 d G ( x ) + 1)((1 − k ) + 4 d G ( x )( k + d G ( x ))) / ,∂ ∂k l ( k, 0) = − d G ( x )( k + 2 d G ( x )) + 4 d G ( x )( k + k − d G ( x )( k + d G ( x ))((1 − k ) + 4 d G ( x )( k + d G ( x ))) / . Since, for all 0 ≤ k ≤ / k − < − k + 1 ≥ ,∂ ∂k l ( k, 0) = − d G ( x )( k + 2 d G ( x )) + 4 d G ( x )( k + k − d G ( x )( k + d G ( x ))( k − k + 1 + 4 d G ( x )( k + d G ( x ))) / ≤ − d G ( x )( k + 2 d G ( x )) + 4 d G ( x )( k + 4 d G ( x )( k + d G ( x ))( k + 4 d G ( x )( k + d G ( x ))) / = − d G ( x )( k + 2 d G ( x )) + 4 d G ( x )( k + 2 d G ( x )) = 0 , the function l ( k, 0) is concave with respect to 0 ≤ k ≤ / 2. It follows from this thatsup ≤ k ≤ / j (0) l ( k, 0) = j (0)min (cid:8) l (0 , , l ( , (cid:9) = max ( , d G ( x )2 p d G ( x ) ) . (3.12)Next, let us consider the latter supremum in (3.11). Since j (2 k − 1) = 1 p d G ( x )( d G ( x ) + 2 k − ,l ( k, k − 1) = kk + 2 d G ( x ) + 1 − k d G ( x ) + 3 k − k + d G ( x ))( k + 2 d G ( x ))(2 d G ( x ) + 3 k − , we will have(3.13) sup / ≤ k ≤ j (2 k − l ( k, k − 1) = sup / ≤ k ≤ ( k + 2 d G ( x ))(2 d G ( x ) + 3 k − k + d G ( x )) p d G ( x )( d G ( x ) + 2 k − . Denote here q ( k ) = k + 2 d G ( x ) p d G ( x )( d G ( x ) + 2 k − 1) and q ( k ) = 2 d G ( x ) + 3 k − k + d G ( x ) . By differentiation, q ′ ( k ) = − d G ( x )( d G ( x ) − k + 1) − d G ( x )( d G ( x ) + 2 k − / = 0 ⇔ k = d G ( x ) + 1 − d G ( x ) . NTRINSIC QUASI-METRICS 9 Since q ( d G ( x ) + 1 − / (4 d G ( x ))) = 3 d G ( x ) + 1 − / (4 d G ( x )) p d G ( x ) + 4 d G ( x ) − p d G ( x ) + 4 d G ( x ) − d G ( x ) ≤ ⇔ p d G ( x ) + 4 d G ( x ) − ≤ d G ( x ) ⇔ − (2 d G ( x ) − ≤ q (1 / 2) = q (1) = 1, the greatest value of the quotient q ( k ) is 1.Similarly, by differentiation and the quadratic formula, q ′ ( k ) = − k + 2(1 − d G ( x )) k + 3 d G ( x )( k + d G ( x )) = 0 ⇔ k = 1 − d G ( x ) ± p d G ( x ) − d G ( x ) + 103 . Because1 − d G ( x ) − p d G ( x ) − d G ( x ) + 103 < − d G ( x ) − | − d G ( x ) | ≤ , the only stationary point on the interval k ∈ (1 / , 1) is k ≡ − d G ( x ) + p d G ( x ) − d G ( x ) + 103 . Thus, the greatest value of the quotient q ( k ) with respect to k is either q ( k ) = 9 p d G ( x ) − d G ( x ) + 10(1 − d G ( x ) + p d G ( x ) − d G ( x ) + 10) + 9 d G ( x ) or q (1 / 2) = q (1) = 2 . By substituting the quotients q ( k ) and q ( k ) in (3.13) with their greatest values foundabove, we will havesup / ≤ k ≤ j (2 k − l ( k, k − ≤ 12 sup / ≤ k ≤ q ( k ) sup / ≤ k ≤ q ( k )= 12 · · max { , p d G ( x ) − d G ( x ) + 10(1 − d G ( x ) + p d G ( x ) − d G ( x ) + 10) + 9 d G ( x ) } = max { , p d G ( x ) − d G ( x ) + 102(1 − d G ( x ) + p d G ( x ) − d G ( x ) + 10) + 18 d G ( x ) } and, by combining this with (3.11) and (3.12),sup ≤ h ≤ , ≤ k ≤ j ( h ) l ( k, h ) = max ( sup ≤ k ≤ / j (0) l ( k, , sup / ≤ k ≤ j (2 k − l ( k, k − ) ≤ max ( , d G ( x )2 p d G ( x ) , p d G ( x ) − d G ( x ) + 102(1 − d G ( x ) + p d G ( x ) − d G ( x ) + 10) + 18 d G ( x ) ) . Clearly 1 + 4 d G ( x )2 p d G ( x ) ≤ √ ⇔ (1 + 4 d G ( x )) ≤ d G ( x ) ) ⇔ − d G ( x ) − d G ( x ) = − − d G ( x )) ≤ d G ( x ) ≤ p d G ( x ) − d G ( x ) + 102(1 − d G ( x ) + p d G ( x ) − d G ( x ) + 10) + 18 d G ( x ) < √ . Consequently, p G ( x, y ) p G ( x, z ) + p G ( z, y ) ≤ sup ≤ h ≤ , ≤ k ≤ j ( h ) l ( k, h ) ≤ √ , which means that p G is a quasi-metric with a constant at most √ / G = ( − , ( R . If x = − / y = 1 / z = 0, we will have p G ( x, y ) = 15 = √ (cid:18) 15 + 15 (cid:19) = √ 52 ( p G ( x, z ) + p G ( z, y )) , so the result follows. (cid:3) Let us now show exactly when the constant of Theorem 3.7 is the best possible, assumingthe domain G is fixed. Proposition 3.14. If there are some k ∈ G ( R n and r > such that B n ( k, r ) ⊂ G and d ( S n − ( k, r ) ∩ ∂G ) = 2 r , then √ / is the best constant with which p G is a quasi-metric.Proof. By Theorem 3.7, p G is always a quasi-metric with a constant at most √ / 2. Fixfirst x ′ , y ′ ∈ S n − ( k, r ) ∩ ∂G so that | x ′ − y ′ | = 2 r and then choose x, y, z so that x ′ , x, z, y, y ′ are collinear, | x ′ − x | = | x − y | = | y ′ − y | = 2 r/ | x − z | = | z − y | = r/ 3. Now, d G ( x ) = | x ′ − x | = 2 r/ | y ′ − y | = d G ( y ) and z = k so d G ( z ) = r . Consequently, p G ( x, y ) = 15 = √ (cid:18) 15 + 15 (cid:19) = √ 52 ( p G ( x, z ) + p G ( z, y )) . (cid:3) An example of a domain where the constant √ / B n . We can easily see this by choosing x = e / z = 0 and y = − e / 3. Otherdomains like this include, for instance, a twice punctured space R n \ ( { s }∪{ t } ), s = t ∈ R n ,and all k -dimensional hypercubes in R n where 1 ≤ k ≤ n . However, note that there arealso domains in which √ / p H n ( x, y ) = s H n ( x, y ), so the point pair function is a metric in a domain G = H n . It alsofollows from this that the point pair function is a metric in a sector with an angle over π . NTRINSIC QUASI-METRICS 11 Theorem 3.15. In an open sector S θ with an angle π ≤ θ < π , the point pair function p S θ is a metric.Proof. Trivially, we only need to prove that the point pair function fulfills the triangleinequality in this domain. Fix distinct points x, y ∈ S θ . Note that if θ ≥ π then, for everypoint x ∈ S θ , there is exactly one point x ′ ∈ S ( x, d S θ ( x )) ∩ ∂S θ . Fix x ′ , y ′ like this for thepoints x, y , respectively. Furthermore, denote the closed hull J = ∪ u ∈ L ( x,x ′ ) , v ∈ L ( y,y ′ ) [ u, v ].The special case where where x, y, x ′ , y ′ are collinear and J is just a line is possible, butthis does not affect our proof.We are interested in such a point z ∈ S θ that minimizes the sum p S θ ( x, z ) + p S θ ( z, y ).This point z must belong in J ∩ S θ : If z / ∈ J , then it can be rotated around either x or y into a new point z ∈ J ∩ S θ so that one of the distances | x − z | and | z − y | does notchange and the other one decreases, and the distance d S θ ( z ) increases.Choose now a half-plane H such that x, y ∈ H and ∂H is a tangent for both S ( x, d S θ ( x ))and S ( y, d S θ ( y )). Now, d H ( x ) = d S θ ( x ) and d H ( y ) = d S θ ( y ). Clearly, J ∩ S θ ⊂ H and,since θ ≥ π , for every point z ∈ J ∩ S θ , d S θ ( z ) ≤ d H ( z ). Recall that the point pair function p G is a metric in a half-plane domain. It follows that p S θ ( x, y )inf z ∈ S θ ( p S θ ( x, z ) + p S θ ( z, y )) = p S θ ( x, y )inf z ∈ J ∩ S θ ( p S θ ( x, z ) + p S θ ( z, y )) ≤ p H ( x, y )inf z ∈ J ∩ S θ ( p H ( x, z ) + p H ( z, y )) ≤ , which proves our result. (cid:3) It can be shown that the point pair function p G is not a metric in a sector S θ withan angle 0 < θ < π . For instance, if θ = π/ 2, then the points x = e π/ , y = e π/ and z = ( x + y ) / Lemma 3.16. For all angles < θ < π , the point pair function p S θ is invariant underthe M¨obius transformation f : S θ → S θ , f ( x ) = x/ | x | . Proof. By Remark 2.3, we can fix x = e ki and y = re hi with r > < k ≤ h < θ without loss of generality. Now, f ( x ) = x = e ki and f ( y ) = (1 /r ) e hi . It follows that p S θ ( x, y ) = | − re ( h − k ) i | p | − re ( h − k ) i | + 4 d S θ ( e ki ) d S θ ( re hi )= r | − (1 /r ) e ( h − k ) i | p r | − (1 /r ) e ( h − k ) i | + 4 r d S θ ( e ki ) d S θ ((1 /r ) e hi )= | − (1 /r ) e ( h − k ) i | p | − (1 /r ) e ( h − k ) i | + 4 d S θ ( e ki ) d S θ ((1 /r ) e hi ) = p S θ ( f ( x ) , f ( y )) , which proves the result. (cid:3) Let us yet consider the connection between the point pair function and the triangularratio metric and, especially, what we can tell about the domain by studying these metrics. Theorem 3.17. [14, Theorem 3.8, p. 5] A domain G ( R n is convex if and only if theinequality s G ( x, y ) ≤ p G ( x, y ) holds for all x, y ∈ G . Theorem 3.18. If G ( R n is a domain such that s G ( x, y ) = p G ( x, y ) holds for all x, y ∈ G , then the following conditions are fulfilled: (1) There is no such line segment [ x, y ] ⊂ R n that x, y ∈ G and [ x, y ] ∩ ∂G = ∅ , (2) there is no such line segment [ u, v ] ⊂ R n that u, v ∈ ∂G and [ u, v ] ∩ G = ∅ .Proof. (1) If the equality s G ( x, y ) = p G ( x, y ) holds for all x, y ∈ G , it follows from Theo-rem 3.17 that G is convex.(2) Fix u, v ∈ ∂G so that [ u, v ] ∩ G = ∅ . Now, there must be some u ′ , v ′ ∈ [ u, v ]such that u ′ , v ′ ∈ ∂G but ( u ′ + v ′ ) / ∈ G . Fix x, y ∈ G so that r = d ( x, [ u ′ , v ′ ]) = d G ( x ) = d ( y, [ u ′ , v ′ ]) = d G ( y ), | x − u ′ | ≤ √ r and | y − v ′ | ≤ √ r with some verysmall distance r > 0. It follows from the first part of this theorem that the convexhull ∪ u ∈ B n ( x,r ) , v ∈ B n ( y,r ) [ u, v ] must belong to G . Suppose that each point q inside thisbounded area fulfills d ( q, [ u ′ , v ′ ]) ≤ d ( q, ∂G ); if not, choose a smaller r .Consider now the quotient s G ( x, y ) p G ( x, y ) = p | x − y | + 4 d G ( x ) d G ( y )inf z ∈ ∂G ( | x − z | + | z − y | ) . (3.19)Here, clearly d G ( x ) = d G ( y ) = r . Since ( u ′ + v ′ ) / ∈ G and no other border points closeto the other side of the segment [ x, y ], it follows thatinf z ∈ ∂G ( | x − z | + | z − y | ) > p | x − y | / r = p | x − y | + 4 r . Thus, the quotient (3.19) is truly smaller than 1 and the equality s G ( x, y ) = p G ( x, y )cannot hold. (cid:3) The next result follows. NTRINSIC QUASI-METRICS 13 Corollary 3.20. If G ( R n is a domain such that s G ( x, y ) = p G ( x, y ) holds for all x, y ∈ G , then G can be transformed into a half-space with only translations and rotations. New quasi-metric In this section, we define a new intrinsic quasi-metric w G in a convex domain G andstudy its basic properties. As can be seen from Theorem 4.5, this function gives a lowerbound for the triangular ratio metric. Since the point pair function serves as an upperbound for the triangular ratio metric, these two quasi-metrics can be used to form boundsfor the triangular ratio distance like in Corollary 4.7. Furthermore, these three functionsare equivalent in the case of the half-space, see Proposition 4.2, so these bounds are clearlyessentially sharp at least in some cases.First, consider the following definition. Definition 4.1. Let G ( R n be a convex domain. For any x ∈ G , there is a non-emptyset e X = { e x ∈ S n − ( x, d G ( x )) | ( x + e x ) / ∈ ∂G } . Define now a function w G : G × G → [0 , w G ( x, y ) = | x − y | min { inf e y ∈ e Y | x − e y | , inf e x ∈ e X | y − e x |} , x, y ∈ G. Note that we can only define the function w G for convex domains G because, for anon-convex domain G and some points x, y ∈ G , there is some points x, y ∈ G such that y = e x with some e x ∈ e X and the denominator in the expression of w G would become zero. Proposition 4.2. For all points x, y ∈ H n , w H n ( x, y ) = s H n ( x, y ) = p H n ( x, y ) . Proof. For all x = ( x , ..., x n ) ∈ H n , there is only one point e x = ( x , ..., x n − , − x n ) = x − x n in the set e X . Thus, w H n ( x, y ) = | x − y | min {| x − e y | , | y − e x |} = | x − y | min {| x − y + 2 y n | , | y − x + 2 x n |} = | x − y | p | x − y | + 4 x n y n = p H n ( x, y ) . The result s H n ( x, y ) = p H n ( x, y ) is in Lemma 2.5(1). (cid:3) While it trivially follows from the result above that the function w G is a metric in thecase G = H n , this is not true for all convex domains G , as can be seen with the followingexample. Example 4.3. G = { z ∈ C | − < Re( z ) < , < Im( z ) < } is a convex domain, inwhich w G is not a metric. Proof. Let x = 1 / k + i/ y = − / i/ z = − / − k + i/ < k < / w G ( x, y ) = 1 + k p k ) , w G ( x, z ) = 1 + 2 k , and w G ( z, y ) = k − k . Since w G ( x, y ) w G ( x, z ) + w G ( z, y ) = 1 + k p k ) ,(cid:18) k k − k (cid:19) = 2(1 − k ) p k ) (1 + 3 k − k )we will havelim k → + w G ( x, y ) w G ( x, z ) + w G ( z, y ) = lim k → + − k ) p k ) (1 + 3 k − k ) = √ . (cid:3) It follows also from Example 4.3 that there cannot be a constant less than √ w G would be a quasi-metric for all domains. In fact, we will prove inCorollary 4.6 that, for an arbitrary convex domain, w G is a quasi-metric with the constant √ 2. However, in order to do this, let us first consider two inequalities. Proposition 4.4. For any convex domain G ( R n and all x, y ∈ G , j ∗ G ( x, y ) ≤ w G ( x, y ) .Proof. By the triangle inequality,min { inf e y ∈ e Y | x − e y | , inf e x ∈ e X | y − e x |} ≤ min {| x − y | + d ( y, e Y ) , | x − y | + d ( x, e X ) } = min {| x − y | + 2 d G ( y ) , | x − y | + 2 d G ( x ) } = | x − y | + 2 min { d G ( x ) , d G ( y ) } , so the result follows. (cid:3) Theorem 4.5. For an arbitrary convex domain G ( R n and all x, y ∈ G , w G ( x, y ) ≤ s G ( x, y ) ≤ √ w G ( x, y ) . Proof. Choose any distinct x, y ∈ G . By symmetry, we can suppose that inf e x ∈ e X | y − e x | ≤ inf e y ∈ e Y | x − e y | . Fix e x as the point that gives this smaller infimum. Let us only considerthe two-dimensional plane containing x, y, e x and set n = 2. Fix u = [ x, e x ] ∩ ∂G and z = [ y, e x ] ∩ ∂G . Position the domain G on the upper half-plane so that the real axis isthe tangent of S ( x, d G ( x )) at the point u . Since G is convex, it must be a subset of H and therefore z ∈ H ∪ R . Thus, it follows that | x − z | ≤ | z − e x | . Consequently, w G ( x, y ) = | x − y || y − e x | = | x − y || z − e x | + | z − y | ≤ | x − y || x − z | + | z − y | ≤ s G ( x, y ) . The inequality s G ( x, y ) ≤ √ w G ( x, y ) follows from Lemma 2.4(3) and Proposition 4.4. (cid:3) NTRINSIC QUASI-METRICS 15 Now, we can show that the function w G is a quasi-metric. Corollary 4.6. For an arbitrary convex domain G ( R n , the function w G is a quasi-metric with a constant greater than or equal to √ , and this is constant is the best possible.Proof. It follows from Theorem 4.5 and the fact that the triangular ratio metric is alwaysa metric that w G ( x, y ) ≤ s G ( x, y ) ≤ s G ( x, z ) + s G ( z, y ) ≤ √ w G ( x, z ) + w G ( z, y ))and, by Example 4.3, the constant √ (cid:3) We will also have the following result. Corollary 4.7. For any convex domain G ( R n and all x, y ∈ G , j ∗ G ( x, y ) ≤ w G ( x, y ) ≤ s G ( x, y ) ≤ p G ( x, y ) . Proof. Follows from Proposition 4.4 and Theorems 4.5 and 3.17. (cid:3) Quasi-metrics in the unit disk In this section, we will focus on the inequalities between the hyperbolic type metricsand quasi-metrics in the case of the unit disk. Calculating the exact value of the triangularratio metric in the unit disk is not a trivial task, but instead quite a difficult problem witha very long history, see [3] for more details. However, we already know from Corollary4.7 that the quasi-metric w G serves as a lower bound for the triangular ratio metric inconvex domains G and this helps us considerably. Remark 5.1. Note that while we focus below just on the unit disk B , all the inequalitiescan be extended to the general case with the unit ball B n , because the values of the metricsand quasi-metrics considered only depend on how the points x, y are located on the two-dimensional place containing them and the origin.First, we will define the function w G of Definition 4.1 in the case G = B . Denote below e x = x (2 − | x | ) / | x | for all points x ∈ B \{ } . We will have the following results. Proposition 5.2. If x, y ∈ B \{ } such that | y | ≤ | x | , then | y − e x | ≤ | x − e y | .Proof. Let µ = ∡ XOY , where o is the origin. Note that | e x | = 2 − | x | and | e y | = 2 − | y | .By the law of cosines, | y − e x | ≤ | x − e y |⇔ p | y | + (2 − | x | ) − | y | (2 − | x | ) cos( µ ) ≤ p | x | + (2 − | y | ) − | x | (2 − | y | ) cos( µ ) ⇔ | y | − (2 − | y | ) − | x | + (2 − | x | ) + 2 | x | (2 − | y | ) cos( µ ) − | y | (2 − | x | ) cos( µ ) ≤ ⇔ | y | + 4 − | x | − | x | (2 − | y | ) − | y | (2 − | x | )) cos( µ ) ≤ ⇔ | y | − | x | ) + 4( | x | − | y | ) cos( µ ) ≤ ⇔ ( | y | − | x | )(1 − cos( µ )) ≤ ⇔ | y | ≤ | x | , which proves the result. (cid:3) Proposition 5.3. If x ∈ B \{ } is fixed and y → , then | x − y || y − e x | → | x | − | x | . Proof. By writing µ = ∡ XOY and using the law of cosines,lim | y |→ + | x − y || y − e x | = lim | y |→ + s | x | + | y | − | x || y | cos( µ ) | y | + (2 − | x | ) − | y | (2 − | x | ) cos( µ ) = s | x | (2 − | x | ) = | x | − | x | . (cid:3) Now, consider the following definition. Definition 5.4. In the domain G = B , the quasi-metric w G is a function w B : B × B → [0 , w B ( x, y ) = | x − y | min {| x − e y | , | y − e x |} , x, y ∈ B \{ } ,w B ( x, 0) = | x | − | x | , x ∈ B , where e x = x (2 − | x | ) / | x | and e y = y (2 − | y | ) / | y | .By Proposition 5.2, for all points x, y ∈ B such that 0 < | y | ≤ | x | < w B ( x, y ) = | x − y || y − e x | . It follows from this and Proposition 5.3 that the function w B of Definition 5.4 is continu-ous. It is also easy to verify that this function is truly equivalent to that of Definition 4.1and, by Corollary 4.6, the function w B truly is at least a quasi-metric. In fact, accordingto the numerical tests, the function w B seems to fulfill the triangle inequality on the unitdisk, which would mean that the following conjecture holds. Conjecture 5.5. The function w B is a metric on the unit disk. However, it does not affect the results of this paper if the function w B truly is a metricor just a quasi-metric, since we use it to create new inequalities between it and somehyperbolic type metrics. Thus, let us move on and show instead that this function w B is quite a good lower bound for the triangular ratio metric in the unit disk. First, beforeproving Theorem 5.7, note that the value of the triangular ratio metric can be calculatedfor the collinear points with the help of the following lemma. Lemma 5.6. [5, 11.2.1(1) p. 205] For all x, y ∈ B n , s B n ( x, y ) ≤ | x − y | − | x + y | , where the equality holds if the points x, y are collinear with the origin. NTRINSIC QUASI-METRICS 17 Theorem 5.7. For all x, y ∈ B , w B ( x, y ) ≤ s B ( x, y ) and the equality holds here when-ever x, y are collinear with the origin.Proof. The inequality follows from Theorem 4.5. Suppose now that the points x, y ∈ B \{ } are collinear with the origin. Without loss of generality, we can fix 0 < y < 1, sothat | x − e y | = | x − (2 − y ) | . Now, by Lemma 5.6, s B ( x, y ) = | x − y | − | x + y | ≤ | x − y || − ( x + y ) | = | x − y || x − (2 − y ) | = | x − y || x − e y | ≤ w B ( x, y ) . (5.8)By combining this inequality (5.8) with the one in Theorem 4.5, we will have the equality s B ( x, y ) = w B ( x, y ) for all points x, y ∈ B \{ } collinear with the origin. In the specialcase where y = 0, s B ( x, 0) = | x | − | x | + 1 = | x | − | x | = w B ( x, , so the theorem follows. (cid:3) By [13, Lemma 3.12, p. 7], the following function can be used to create a lower boundfor the triangular ratio metric. Definition 5.9. [13, Def. 3.9, p. 7] For x, y ∈ B \{ } , definelow( x, y ) = | x − y | min {| x − y ∗ | , | x ∗ − y |} , where x ∗ = x/ | x | and y ∗ = y/ | y | .However, the quasi-metric w B is a better lower bound for the triangular ratio metricin the unit disk than this low-function. Lemma 5.10. For all distinct points x, y ∈ B \{ } , w B ( x, y ) > low( x, y ) .Proof. Let x ∈ B , k > µ = ∡ ([ x, , [1 , k ]). Now, | x − k | = p | x − | + ( k − − | x − | ( k − 1) cos( µ ) . Since π/ < µ ≤ π , cos( µ ) < 0. Thus, we see that the distance | x − k | is strictly increasingwith respect to k . In other words, the further away a point k ∈ R \ B is from the origin,the longer the distance between k and an arbitrary point x ∈ B is. For every point y ∈ B \{ } ,1 < − | y | < | y | ⇔ < (cid:12)(cid:12)(cid:12)(cid:12) y (2 − | y | ) | y | (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) y | y | (cid:12)(cid:12)(cid:12)(cid:12) ⇔ < | e y | < | y ∗ | , so it follows by the observation made above that, for all x ∈ B , | x − e y | < | x − y ∗ | . Consequently, by symmetry, the inequality w B ( x, y ) = | x − y | min {| x − e y | , | y − e x |} > | x − y | min {| x − y ∗ | , | y − x ∗ |} = low( x, y )holds for all points x, y ∈ B \{ } . (cid:3) Next, we will prove one sharp inequality between the two quasi-metrics considered thispaper. Theorem 5.11. For all points x, y ∈ B , w B ( x, y ) ≤ p B ( x, y ) ≤ √ w B ( x, y ) , where the equality w B ( x, y ) = p B ( x, y ) holds whenever x, y are on the same radius, and p B ( x, y ) = √ w B ( x, y ) holds when x = − y and | x | = | y | = 1 / .Proof. For all points x, y ∈ B , the inequality w B ( x, y ) ≤ p B ( x, y ) follows from Corollary4.7. Let us now prove that the equality in the case where x, y are on the same radius. If0 < y ≤ x < w B ( x, y ) = x − y − x − y = x − y p ( x − y ) + 4(1 − x )(1 − y ) = p B ( x, y )and, in the special case y = 0, w B ( x, 0) = | x | − | x | = | x | p | x | + 4(1 − | x | ) = p B ( x, . (5.12)Next, let us prove the latter part of the inequality. We need to find that the maximumof the quotient p B ( x, y ) w B ( x, y ) = min {| x − e y | , | y − e x |} p | x − y | + 4(1 − | x | )(1 − | y | ) . (5.13)In order to do that, we can suppose without loss of generality that x, y are on differentradii since, as we proved above, the equality w B ( x, y ) = p B ( x, y ) holds otherwise. Choosethese points so that 0 < | y | ≤ | x | < µ = ∡ XOY . It follows from Proposition 5.2that the quotient (5.13) is now p B ( x, y ) w B ( x, y ) = | y − e x | p | x − y | + 4(1 − | x | )(1 − | y | )= s | y | + (2 − | x | ) − | y | (2 − | x | ) cos( µ ) | x | + | y | − | x || y | cos( µ ) + 4 − | x | − | y | + 4 | x || y | = s | y | + (2 − | x | ) − | y | (2 − | x | ) cos( µ ) | y | + (2 − | x | ) − | y | (1 − | x | ) − | x || y | cos( µ ) . (5.14) NTRINSIC QUASI-METRICS 19 Fix now j = cos( µ ) , s = | y | + (2 − | x | ) , t = 2 | y | (2 − | x | ) ,u = | y | + (2 − | x | ) − | y | (1 − | x | ) , v = 2 | x || y | , so that the quotient that is the argument of the square root in the expression (5.14) canbe described with a function f : [0 , → R , f ( j ) = s − tju − vj . By differentiation, the function f is decreasing with respect to j , if f ′ ( j ) = − t ( u − vj ) + v ( s − tj )( u − vj ) = sv − tu ( u − vj ) ≤ ⇔ sv − tu ≤ . Since this last inequality is equivalent to( | y | + (2 − | x | ) )2 | x || y | − | y | (2 − | x | )( | y | + (2 − | x | ) − | y | (1 − | x | )) ≤ ⇔ ( | y | + (2 − | x | ) ) | x | − (2 − | x | )( | y | + (2 − | x | ) − | y | (1 − | x | )) ≤ ⇔ | y | | x | + | x | (2 − | x | ) − | y | (2 − | x | ) − (2 − | x | ) + 4 | y | (1 − | x | )(2 − | x | ) ≤ ⇔ − | y | (1 − | x | ) − − | x | )(1 − | x | ) + 4 | y | (1 − | x | )(2 − | x | ) ≤ ⇔ | y | + (2 − | x | ) − | y | (2 − | x | ) ≥ ⇔ ( | y | − (2 − | x | )) = (2 − | x | − | y | ) ≥ , which clearly holds, it follows that the function f and the quotient (5.13) are decreasingwith respect to j = cos( µ ). The minimum value of cos( µ ) is − µ = π . Consequently,we can fix the points x, y so that x = h and y = − h + k with 0 ≤ k < h < 1, without lossof generality. Now, the quotient (5.13) is p B ( x, y ) w B ( x, y ) = 2 − k p (2 h + k ) + 4(1 − h )(1 − h + k ) = r − k + k h − h + k + 4 k + 4 ≤ s − k + k + k (4 − k )8 h − h + k + 4 k + 4 − k (4 + k ) = r h − h + 4 = 1 √ h − h + 1 . This upper bound found above is the value of the quotient (5.13) in the case k = 0,because, for x = h and y = − h , p B ( x, y ) w B ( x, y ) = 2 p h + 4(1 − h ) = 1 √ h − h + 1 . The expression 2 h − h + 1 obtains its minimum value 1 / h = 1 / 2, so the maximumvalue of the quotient (5.13) is √ (cid:3) The next result follows. Theorem 5.15. For all x, y ∈ B , j ∗ B ( x, y ) ≤ w B ( x, y ) ≤ √ j ∗ B ( x, y ) , where the equality j ∗ B ( x, y ) = w B ( x, y ) holds whenever x, y are on the same radius andthe inequality w B ( x, y ) ≤ √ j ∗ B ( x, y ) has the best constant possible.Proof. The inequality j ∗ B ( x, y ) ≤ w B ( x, y ) follows from Proposition 4.4 and the inequality w B ( x, y ) ≤ √ j ∗ B ( x, y ) from Lemma 2.4(1) and Theorem 5.11. Consider now the casewhere x, y are on the same radius, but neither of them is the origin. Without loss ofgenerality, we can suppose that 0 < y ≤ x < w B ( x, y ) = x − y − x − y = x − yx − y + 2(1 − x ) = j ∗ B ( x, y ) . If y = 0 instead, then w B ( x, 0) = | x | − | x | = | x || x | + 2(1 − | x | ) = j ∗ B ( x, y ) . Next, let us show that the constant √ x = 1 − k and y = (1 − k ) e ki for some 0 < k < 1. Now, w B ( x, y ) j ∗ B ( x, y ) = | (1 − k ) − (1 − k ) e ki | + 2(1 − (1 − k )) | (1 − k ) − (2 − (1 − k )) e ki | = 2(1 − k ) sin( k/ 2) + 2 k p (1 − k ) + (1 + k ) − − k ) cos( k ) = √ − k ) sin( k/ 2) + k ) p k − (1 − k ) cos( k ) . Since lim k → − w B ( x, y ) j ∗ B ( x, y ) = lim k → − √ − k ) sin( k/ 2) + k ) p k − (1 − k ) cos( k ) = √ , the result follows. (cid:3) Let us now focus on how the quasi-metric w B can be used to create an upper boundfor the triangular ratio metric. We know from Theorem 4.5 that in the general case wherethe domain G is convex, the inequality s G ( x, y ) ≤ √ w G ( x, y ) holds. Thus, this mustalso hold in the unit disk, but several numerical tests suggest that the constant √ G = B . The next result tells the best constant in acertain special case. Lemma 5.16. For all x, y ∈ B such that | x | = | y | and ∡ XOY = π/ , s B ( x, y ) ≤ c · w B ( x, y ) with c = s h − h + 22 h − √ h + 2 , h = 1 − p − √ − √ . NTRINSIC QUASI-METRICS 21 Proof. Let x = h and y = hi for 0 < h < 1. Now, | h − e πi/ | > ⇔ | √ h − − i | > √ ⇔ q h − √ h + 2 > √ ⇔ h − √ h > ⇔ h > √ . Then s B ( x, y ) = h, if h > √ s B ( x, y ) = h/ √ q (1 − h/ √ + h / h p h − √ h + 2 otherwise, w B ( x, y ) = √ h | hi − (2 − h ) | = √ h √ h − h + 4 = h √ h − h + 2 . If h > / √ s B ( x, y ) w B ( x, y ) = √ h − h + 2 < q (1 / √ − / √ 2) + 2 = q / − √ ≈ . . (5.17)If h ≤ / √ s B ( x, y ) w B ( x, y ) = s h − h + 22 h − √ h + 2 . (5.18)Let f : (0 , / √ → R , f ( h ) = h − h + 22 h − √ h + 2 . By differentiation, f ′ ( h ) = (2 h − h − √ h + 2) − (4 h − √ h − h + 2)(2 h − √ h + 2) = 2((2 − √ h − h + 2 √ − h − √ h + 2) . By the quadratic formula, f ′ ( h ) = 0 holds when h = 2 ± q − − √ √ − − √ 2) = 1 ± p − √ − √ . Here, the ± -symbol must be minus, so that 0 < h ≤ / 2. Fix h = 1 − p − √ − √ ≈ . . Since f ′ (0 . > f ′ (0 . < 1, the function f obtains its local maximum of the interval(0 , / √ 2] at h . Thus, p f ( h ) is the maximum value of the quotient (5.18) within thelimitation h ≤ / √ 2. Since p f ( h ) = s h − h + 22 h − √ h + 2 ≈ . h > / √ w B in the general case. Thus, the lemma follows. (cid:3) The special choice of points x, y in Lemma 5.16 is relevant because several numericaltests suggest that the inequality of this lemma holds more generally. Conjecture 5.20. For all x, y ∈ B , the inequality s B ( x, y ) ≤ c · w B ( x, y ) holds with thesharp constant c = s h − h + 22 h − √ h + 2 ≈ . , h = 1 − p − √ − √ . Consequently, the quasi-metric w B is quite a good estimate for the triangular ratiometric in the unit disk. Especially, if we estimate the triangular ratio distance with thevalue of ( c/ · w B ( x, y ), where c is like above, our error is always less than 3 . w G . Corollary 5.21. [5, p. 460] For all x, y ∈ G ∈ { H n , B n } , (1) th ρ H n ( x, y )4 ≤ j ∗ H n ( x, y ) ≤ w H n ( x, y ) = s H n ( x, y ) = p H n ( x, y ) = th ρ H n ( x, y )2 , (2) th ρ B n ( x, y )4 ≤ j ∗ B n ( x, y ) ≤ w B n ( x, y ) ≤ s B n ( x, y ) ≤ p B n ( x, y ) ≤ th ρ B n ( x, y )2 . Proof. Follows from Lemma 2.5, Proposition 4.2, Theorems 5.7 and 5.15, and Remark5.1. (cid:3) References A.F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Proc. In-ternational Workshop on Quasiconformal Mappings and their Applications (IWQCMA05), eds. 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Vuorinen, Triangular Ratio Metric Under Quasiconformal Mappings In SectorDomains. Arxiv, 2005.11990. (2020). Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Fin-land Email address ::