A Gromov Hyperbolic metric and Möbius transformations
aa r X i v : . [ m a t h . C V ] J un A GROMOV HYPERBOLIC METRIC AND M ¨OBIUSTRANSFORMATIONS
XIAOXUE XU, GENDI WANG*, AND XIAOHUI ZHANG
Abstract.
We compare a Gromov hyperbolic metric with the hyperbolic metric in theunit ball or in the upper half space, and prove sharp comparison inequalities betweenthe Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain severalsharp distortion inequalities for the Gromov hyperbolic metric under some families ofM¨obius transformations.
Keywords.
Gromov hyperbolic metric, hyperbolic metric, hyperbolic type metrics,M¨obius transformations
Introduction
It is well known that metrics play important roles in geometric function theory. One ofthe most important metrics is the hyperbolic metric in the unit ball or in the upper halfspace. In addition to the classical hyperbolic metric, numerous hyperbolic type metricsare natural generalizations of the hyperbolic metric. The most important property of thehyperbolic metric is its invariance under a group of M¨obius transformations. Examples ofM¨obius invariant metrics also include the Seittenranta metric [S], the Apollonian metric[B2], and the M¨obius invariant Cassinian metric [I3]. In order to better understand thesemetrics, various estimates between the hyperbolic metric and hyperbolic type metrics areinvestigated [AVV, CHKV, HL, H2, IMSZ, S, V, Z].The most used hyperbolic type metrics are the quasihyperbolic metric and the distanceratio metric [GP, GO]. Whereas both metrics are not M¨obius invariant, then it is naturalto study the quasi-invariance properties for these metrics. Namely, it would be interestingto obtain the Lipschitz constants for these metrics under M¨obius transformations. Indeed,Gehring, Palka, and Osgood have proved that the quasihyperbolic metric and the distanceratio metric are not changed by more than a constant 2 under M¨obius transformations, see[GP, Corollary 2.5] and [GO, proof of Theorem 4]. Several authors have also studied thistopic for other hyperbolic type metrics in [CHKV, HVZ, I3, KLVW, MS1, MS2, SVW,WV, XW].Recently, Ibragimov introduced a new metric u Z to hyperbolize the locally compactnoncomplete metric space ( Z, d ) without changing its quasiconformal geometry which isdefined as [I2] u Z ( x, y ) = 2 log d ( x, y ) + max { d ( x, ∂Z ) , d ( y, ∂Z ) } p d ( x, ∂Z ) d ( y, ∂Z ) , x, y ∈ Z , where d ( x, ∂Z ) is the distance from the point x to the boundary of Z . For a domain D ( R n equipped with the Euclidean metric, we have [MS1] u D ( x, y ) = 2 log | x − y | + max { d ( x ) , d ( y ) } p d ( x ) d ( y ) , x, y ∈ D, File: Gromov20200607.tex, printed: 2020-6-9, 0.46 where d ( x ) denotes the Euclidean distance from x to the boundary of D .Several authors have studied comparison inequalities between the Gromov hyperbolicmetric and the hyperbolic metric as well as some hyperbolic type metrics [I2, MS1, Z].Mohapatra and Sahoo also investigated quasi-invariance properties of the Gromov hyper-bolic metric under quasiconformal mappings [MS2].In this paper, we continue the investigation on the Gromov hyperbolic metric to improveor complement some results in [MS1]. We further obtain sharp comparison inequalitiesbetween the Gromov hyperbolic metric and the hyperbolic metric, the distance ratiometric, and some other related hyperbolic type metrics such as the Seittenranta metric,the half-Apollonian metric and so on. We also prove sharp distortion inequalities for theGromov hyperbolic metric under some specific families of M¨obius transformations.2. Preliminaries
In this section, for readers’ convenience, we collect the definitions and some basic prop-erties of various hyperbolic type metrics.
The hyperbolic metrics ρ B n and ρ H n of the unit ball B n = { z ∈ R n : | z | < } and of the upper half space H n = { x = ( x , . . . , x n ) ∈ R n : x n > } aredefined as follows. By [B1, p.40], for x, y ∈ B n ,sh ρ B n ( x, y )2 = | x − y | p (1 − | x | )(1 − | y | ) , and hence [Z, (3)] ρ B n ( x, y ) = 2 log p | x − y | + (1 − | x | )(1 − | y | ) + | x − y | p (1 − | x | )(1 − | y | ) . (2.2)By [B1, p.35], for x, y ∈ H n , ch ρ H n ( x, y ) = 1 + | x − y | x n y n , and hence [Z, (8)] ρ H n ( x, y ) = log | x − y | + p | x − y | + 4 x n y n | x − y | x n y n ! . (2.3)Two special formulas of the hyperbolic metric are frequently used [V, (2.17),(2.6)] : ρ B n ( re , se ) = log (cid:18) s − s · − r r (cid:19) , for − < r < s < , s > , and ρ H n ( re n , se n ) = log sr , for 0 < r < s . The following lemma shows the relation between the metric u D and the metric ρ D when D ∈ { B n , H n } . Theorem 2.4. [Z, Theorem 1, Theorem 2] , [MS1, Theorem 3.6] ρ B n ( x, y ) ≤ u B n ( x, y ) ≤ ρ B n ( x, y ) , f or x, y ∈ B n .ρ H n ( x, y ) ≤ u H n ( x, y ) ≤ ρ H n ( x, y ) , f or x, y ∈ H n . All the inequalities are sharp.
GROMOV HYPERBOLIC METRIC AND M ¨OBIUS TRANSFORMATIONS 3
In [GO, p.51], Gehring and Osgood introduced the distanceratio metric ˜ j D . Let D be a proper open subset of R n . For x, y ∈ D ,˜ j D ( x, y ) = 12 log (cid:18) | x − y | d ( x ) (cid:19) (cid:18) | x − y | d ( y ) (cid:19) . Vuorinen made some modification of the above definition and defined the metric j D ,still called the distance ratio metric, as follows [V, (2.34)]: j D ( x, y ) = log (cid:18) | x − y | min { d ( x ) , d ( y ) } (cid:19) . The following lemma shows the relation between the distance ratio metric and thehyperbolic metric.
Lemma 2.6. [AVV, Lemma 7.56] , [V, Lemma 2.41(2)]12 ρ B n ( x, y ) ≤ j B n ( x, y ) ≤ ρ B n ( x, y ) , f or x, y ∈ B n . ρ H n ( x, y ) ≤ j H n ( x, y ) ≤ ρ H n ( x, y ) , f or x, y ∈ H n . Theorem 2.7. [MS1, Lemma 3.1]
Let D ( R n be arbitrary. Then j D ( x, y ) ≤ u D ( x, y ) ≤ j D ( x, y ) . The first inequality becomes equality when d ( x ) = d ( y ) . Theorem 2.8. [MS1, Theorem 4.8]
For D ( R n , we have j D ( x, y ) ≤ u D ( x, y ) ≤ j D ( x, y ) . The first inequality is sharp.
For an open subset D of R n with card( ∂D ) ≥ x, y ∈ D , the Seittenranta metric δ D is defined as [S] δ D ( x, y ) = log (cid:18) p,q ∈ ∂D | p, x, q, y | (cid:19) , where | p, x, q, y | = | p − q | | x − y || p − x | | q − y | with |∞ − q ||∞ − x | = 1is the absolute ratio.The most important property of the absolute ratio is its invariance under M¨obius trans-formations [B1, Theorem 3.2.7]. It follows from the definitions that [S, Remarks 3.2(3)] δ R n \{ ζ } ( x, y ) = j R n \{ ζ } ( x, y )for all ζ ∈ R n .The distance ratio metric and the Seittenranta metric are comparable as the followinglemma shows. Lemma 2.10. [S, Theorem 3.4]
The inequalities j D ( x, y ) ≤ δ D ( x, y ) ≤ j D ( x, y ) ≤ j D ( x, y ) hold for every open set D ( R n . XIAOXUE XU, GENDI WANG*, AND XIAOHUI ZHANG
For a proper open subset D of R n and for all x, y ∈ D , theApollonian metric α D is defined as [B2] α D ( x, y ) = sup p,q ∈ ∂D log | p, x, y, q | . Note that α D is a pseudo-metric in D . It is, in fact, a metric if and only if R n \ D isnot contained in an ( n − R n [B2, Theorem 1.1]. By [V, Lemma8.39] and [B2, Example 3.2, Lemma 3.1], we have δ D ( x, y ) = α D ( x, y ) = ρ D ( x, y )when D ∈ { B n , H n } .The following lemma shows the relation between the metric α D and the metric j D . Lemma 2.12. [S, Theorem 4.2]
Let D ( R n be a convex domain. Then α D ( x, y ) ≤ j D ( x, y ) . The following lemma shows the relation between the metric δ D and the metric α D . Lemma 2.13. [S, Theorem 3.11]
Let D ⊂ R n be an open set with card( ∂D ) ≥ . Then α D ( x, y ) ≤ δ D ( x, y ) ≤ log (cid:0) e α D ( x,y ) + 2 (cid:1) ≤ α D ( x, y ) + log 3 . The first two inequalities give the best possible bounds for δ D expressed in terms of α D only. In [HL], H¨ast¨o and Lind´en gave another form of the Apollonian metric :(2.14) α D ( x, y ) = sup p ∈ ∂D log | p − y || p − x | + sup q ∈ ∂D log | q − x || q − y | . The half-Apollonian metric is defined by using one term in the right-hand side of (2.14).
For a proper open subset D of R n and for all x, y ∈ D ,the half-Apollonian metric η D is defined as [HL] η D ( x, y ) = sup p ∈ ∂D (cid:12)(cid:12)(cid:12)(cid:12) log | x − p || y − p | (cid:12)(cid:12)(cid:12)(cid:12) . Note that η D is also a pseudo-metric in D , and a proper metric whenever R n \ D is nota subset of a hyperplane [HL, Theorem 1.2]. By [B2, Lemma 2.2 (i)], we have α D ( x, y ) = η D ( x, y )when D = R n \ { ζ , ∞} for any ζ ∈ R n .The following lemma shows the relation between the metric η D and the metric α D . Lemma 2.16. [HL, Theorem 2.1]
Let D ( R n be a domain. Then the double inequality α D ( x, y ) ≤ η D ( x, y ) ≤ α D ( x, y ) holds for all x, y ∈ D . Both inequalities are sharp. For a proper subdomain D of R n and for all x, y ∈ D , theCassinian metric c D is defined as [I1] c D ( x, y ) = sup p ∈ ∂D | x − y || x − p || y − p | . GROMOV HYPERBOLIC METRIC AND M ¨OBIUS TRANSFORMATIONS 5
For a proper subdomain D of R n and for all x, y ∈ D ,the triangular ratio metric s D is defined as [CHKV] s D ( x, y ) = sup p ∈ ∂D | x − y || x − p | + | y − p | . The following lemma shows the relation between the metric s D and the metric j D . Lemma 2.19. [HVZ, Lemma 2.1]
Let D be a proper subdomain of R n . Then th j D ( x, y )2 ≤ s D ( x, y ) ≤ e j D ( x,y ) − . The metric u D and the hyperbolic metric We devote this section to improving the right-hand side of inequalities (3.2) and showthe analogue result in the upper half space.
Theorem 3.1. [MS1, Theorem 3.5]
For all x, y ∈ B n , we have ρ B n ( x, y ) − ≤ u B n ( x, y ) ≤ ρ B n ( x, y ) + 2 log 2 . (3.2) Theorem 3.3.
For all x, y ∈ B n , we have u B n ( x, y ) ≤ ρ B n ( x, y ) + 2 log 2 , (3.4) and the inequality is sharp.Proof. Without loss of generality, we may assume that | x | ≤ | y | < | x − y | + 1 − | x | p (1 − | x | )(1 − | y | ) ≤ | x − y | + p | x − y | + (1 − | x | )(1 − | y | ) p (1 − | x | )(1 − | y | ) , which is equivalent to( | x − y | + 1 − | x | ) p (1 + | x | )(1 + | y | ) ≤ (cid:16) | x − y | + p | x − y | + (1 − | x | )(1 − | y | ) (cid:17) . The above inequality follows from p (1 + | x | )(1 + | y | ) < − | x | ) ≤ (1 − | x || y | ) = ( | x | − | y | ) + (1 − | x | )(1 − | y | ) ≤ | x − y | + (1 − | x | )(1 − | y | ) . This proves the desired inequality.For the sharpness, we set x = se and y = te with 0 < s < t <
1. By (2.2), we havelim t → − ( u B n ( x, y ) − ρ B n ( x, y )) = lim t → − t − s + 1 − s ) p (1 + s )(1 + t ) t − s + p ( t − s ) + (1 − s )(1 − t )= 2 log p s ) . (3.5) XIAOXUE XU, GENDI WANG*, AND XIAOHUI ZHANG
For arbitrary ǫ >
0, there exists 0 < s < p s ) > − ǫ/
2. Itfollows from (3.5) that there exists a number t with s < t < t − s + 1 − s ) p (1 + s )(1 + t ) t − s + p ( t − s ) + (1 − s )(1 − t ) > p s ) − ǫ/ > − ǫ. Hence for arbitrary ǫ >
0, there exist x = s e and y = t e such that u B n ( x, y ) − ρ B n ( x, y ) > − ǫ, which implies the sharpness of the inequality (3.4). (cid:3) Theorem 3.6.
For all x, y ∈ H n , we have u H n ( x, y ) ≤ ρ H n ( x, y ) + 2 log 2 , and the inequality is sharp.Proof. Without loss of generality, we may assume that 0 < y n ≤ x n . By (2.3), it sufficesto prove that (cid:18) | x − y | + x n √ x n y n (cid:19) ≤ | x − y | + | x − y | p | x − y | + 4 x n y n x n y n ! , which is equivalent to x n + 2 | x − y | x n ≤ x n y n + | x − y | + 2 | x − y | p | x − y | + 4 x n y n . The above inequality follows from x n + 2 | x − y | x n ≤ ( | x − y | + y n ) + 2 | x − y | x n ≤ x n y n + | x − y | + 2 | x − y | ( x n + y n )and ( x n + y n ) ≤ | x − y | + 4 x n y n . For the sharpness, we choose x = e n and y = te n with 0 < t < . Thenlim t → + ( u H n ( x, y ) − ρ H n ( x, y )) = lim t → + (cid:18) − t √ t − log 1 t (cid:19) = 2 log 2 . Thus completes the proof. (cid:3) The metric u D and the distance ratio metric Mohapatra and Sahoo [MS1] compared the metric u D with the distance ratio metrics˜ j D and j D . By Theorem 2.7 and Lemma 2.10, we have j D ( x, y ) ≤ j D ( x, y ) ≤ u D ( x, y ) ≤ j D ( x, y ) ≤ j D ( x, y ) , (4.1)and the first two inequalities give the best possible bounds for the metric u D in terms ofthe metrics ˜ j D and j D , see Theorem 2.7 and Theorem 2.8.In this section, we will refine the last two inequalities in (4.1). Specifically, the followingTheorem 4.2 and Corollary 4.4 show that the constant 4 in inequalities (4.1) can beimproved to 3 and 3 is the best possible. GROMOV HYPERBOLIC METRIC AND M ¨OBIUS TRANSFORMATIONS 7
Theorem 4.2.
For all x, y ∈ D ( R n , we have u D ( x, y ) ≤ j D ( x, y ) , (4.3) and the inequality is sharp.Proof. Without loss of generality, we may assume that d ( y ) ≤ d ( x ) . To show our claim,it suffices to prove that | x − y | + d ( x ) p d ( x ) d ( y ) ! ≤ (cid:18) | x − y | d ( x ) (cid:19) (cid:18) | x − y | d ( y ) (cid:19) , which is equivalent to | x − y | d ( x ) + d ( x ) ≤ | x − y | d ( y ) + 3 | x − y | + 3 | x − y | d ( y ) + d ( y ) . The above inequality follows from | x − y | d ( x ) ≤ | x − y | ( | x − y | + d ( y )) = | x − y | + | x − y | d ( y )and d ( x ) ≤ ( | x − y | + d ( y )) = | x − y | + 2 | x − y | d ( y ) + d ( y ) . For the sharpness, we consider the domain D = R n \ { e } . Let x = − y = t e with0 < t < t → + u D ( x, y )˜ j D ( x, y ) = lim t → + t √ − t log (cid:0) t − t (cid:1) (cid:0) t t (cid:1) = 3 . Thus completes the proof. (cid:3)
Corollary 4.4.
For all x, y ∈ D ( R n , we have u D ( x, y ) ≤ j D ( x, y ) , (4.5) and the inequality is sharp.Proof. The inequality follows from Theorem 4.2 and Lemma 2.10 .For the sharpness, we consider the domain D = R n \ { e } . Let x = − y = t e with0 < t < t → + u D ( x, y ) j D ( x, y ) = lim t → + t √ − t log (cid:0) t − t (cid:1) = 3 . Thus completes the proof. (cid:3)
Remark 4.6.
By Lemma 2.6, it is obvious that Corollary 4.4 improves the upper boundsof the metric u D in [Z, Theorem 1, Theorem 2] and [MS1, Theorem 3.6], see Theorem 2.4.Moreover, the inequalities (4.3) and (4.5) hold in arbitrary proper subdomains of R n Theorem 4.7.
For all x, y ∈ D ( R n , we have u D ( x, y ) ≤ j D ( x, y ) + log 2 , and the inequality is sharp. XIAOXUE XU, GENDI WANG*, AND XIAOHUI ZHANG
Proof.
Without loss of generality, we may assume that d ( y ) ≤ d ( x ) . To show our claim,it suffices to prove that( | x − y | + d ( x )) d ( x ) d ( y ) ≤ (cid:18) | x − y | d ( x ) (cid:19) (cid:18) | x − y | d ( y ) (cid:19) , or, equivalently d ( x ) ≤ | x − y | + 2 d ( y ) , which is true by the triangle inequality.For the sharpness, we consider the domain D = R n \ { e } . Let x = − y = t e with0 < t < t → − (cid:0) u D ( x, y ) − j D ( x, y ) (cid:1) = lim t → − (cid:18) t √ − t − log (cid:18) t − t (cid:19) (cid:18) t t (cid:19)(cid:19) = log 2 . Thus completes the proof. (cid:3)
Theorem 4.7 and Lemma 2.10 together yield the following corollary.
Corollary 4.8.
For all x, y ∈ D ( R n , we have u D ( x, y ) ≤ j D ( x, y ) + log 2 . The metric u D and other related metrics In this section, we compare the metric u D with the metrics δ D , η D , α D , c D , and s D ,respectively. Theorem 5.1. [MS1, Corollary 5.4]
For all x, y ∈ D ( R n , we have δ D ( x, y ) ≤ u D ( x, y ) ≤ δ D ( x, y ) . The following theorem is an improvement of Theorem 5.1 and of importance in studyingthe distortion property of the metric u D under M¨obius transformations. Theorem 5.2.
For all x, y ∈ D ( R n , we have δ D ( x, y ) ≤ u D ( x, y ) ≤ δ D ( x, y ) , and both inequalities are sharp.Proof. The inequalities follow from Theorem 2.7, Lemma 2.10 and Corollary 4.4.For the sharpness of the left-hand side of the inequalities, we consider the domain D = B n . Let x = − y = t e with 0 < t < u D ( x, y ) = δ D ( x, y ) = 2 log 1 + t − t . For the sharpness of the right-hand side of the inequalities, we consider the domain D = R n \ { e } , then δ D ( x, y ) = j D ( x, y ) . By Corollary 4.4, the constant 3 is the bestpossible. (cid:3)
Theorem 5.3. [MS1, Lemma 5.14]
Let D ( R n and x, y ∈ D . Then η D ( x, y ) ≤ u D ( x, y ) ≤ (cid:0) e η D ( x,y ) (cid:1) . GROMOV HYPERBOLIC METRIC AND M ¨OBIUS TRANSFORMATIONS 9
Theorem 5.4.
For all x, y ∈ D ( R n , we have η D ( x, y ) ≤ u D ( x, y ) ≤ η D ( x, y ) + 2 log 3 . The constant in the left-hand side and the constant in the right-hand side of theinequalities are the best possible.Proof. The left-hand side of the inequalities is the fact of Theorem 5.3.For the sharpness of the constant 1 , we consider the domain D = R n \ { e } . Let x = 0and y = t e with 0 < t < η D ( x, y ) = log 11 − t and u D ( x, y ) = 2 log 1 + t √ − t . Now we see that lim t → − u D ( x, y ) η D ( x, y ) = lim t → − t √ − t log − t = 1 . To prove the right-hand side of the inequalities, we assume that d ( x ) ≤ d ( y ) . Choose z ∈ ∂D such that | x − z | = d ( x ) . This implies | x − z | ≤ | y − z | and | x − y | ≤ | y − z | . Then u D ( x, y ) = 2 log | x − y | + d ( y ) p d ( x ) d ( y ) ! ≤ (cid:18) | x − y | + d ( y ) | x − z | (cid:19) ≤ (cid:18) | y − z || x − z | (cid:19) ≤ w ∈ ∂D log (cid:18) | y − w || x − w | (cid:19) = 2 η D ( x, y ) + 2 log 3 . For the sharpness of the constant 2 log 3 , let D = R n \{ } and y = − x . Then u D ( x, y ) =2 log 3 and η D ( x, y ) = 0.Thus completes the proof. (cid:3) Remark 5.5.
Let f ( t ) = 4 log (cid:0) e t (cid:1) − t − , where t = η D ( x, y ) ∈ [0 , + ∞ ) . By differentiation, we have f ′ ( t ) = 2 e t −
42 + e t , which is negative on (0 , log 2) and positive on (log 2 , + ∞ ). Hence, we have f min ( t ) = f (log 2) = 6 log 2 − > . Therefore, the right-hand side of the inequalities in Theorem 5.4 is better than that inTheorem 5.3.
Theorem 5.6.
For all x, y ∈ D ( R n , we have α D ( x, y ) ≤ u D ( x, y ) ≤ α D ( x, y ) + 2 log 3 . The constant in the left-hand side and the constant in the right-hand side of theinequalities are the best possible. Proof.
The left-hand side of the inequalities follows from Theorem 5.2 and Lemma 2.13.For the sharpness of the constant 1 , we consider the domain D = R n \ { e } , then α D ( x, y ) = η D ( x, y ) . The result follows from Theorem 5.4.The right-hand side of the inequalities follows from Theorem 5.4 and Lemma 2.16.For the sharpness of the constant 2 log 3 , we consider the domain D = R n \ { } , then α D ( x, y ) = η D ( x, y ) . The result follows from Theorem 5.4. (cid:3) Theorem 5.7.
Let D ( R n be a convex domain. Then for all x, y ∈ D , u D ( x, y ) ≤ α D ( x, y ) , and the inequality is sharp.Proof. The inequality follows from Corollary 4.4 and Lemma 2.12.By Theorem 2.4, the constant 3 is the best possible since α D ( x, y ) = ρ D ( x, y ) for D = B n . (cid:3) Theorem 5.8.
For all x, y ∈ D ( R n , we have u D ( x, y ) ≥ r c D ( x, y )) , where r = min { d ( x ) , d ( y ) } . The inequality is sharp.Proof. The inequality follows from [XW, Theorem 4] and [MS1, Theorem 4.5].For the sharpness, we consider the punctured space D p = R n \ { p } . Let x, y ∈ D p with | x − p | = | y − p | . It is clear that u D ( x, y ) = 2 log (cid:18) | x − y || x − p | (cid:19) = 2 log (1 + | x − p | c D ( x, y )) . Hence the inequality is sharp. (cid:3)
Remark 5.9.
By [MS1, Corollary 5.6], we have u D ( x, y ) ≥
12 log (1 +
R c D ( x, y )) , where R = max { d ( x ) , d ( y ) } . It is easy to see that the result in Theorem 5.8 is betterthan that in [MS1, Corollary 5.6] when r ≥ R . Theorem 5.10.
For all x, y ∈ D ( R n , we have (2 log 3) s D ( x, y ) ≤ u D ( x, y ) ≤ s D ( x, y )1 − s D ( x, y ) , and both inequalities are sharp.Proof. The left-hand side of the inequalities is the fact of [MS1, Corollary 5.10].For the sharpness of the left-hand side of the inequalities, we consider the domain D = R n \ { } . Setting y = − x , then s D ( x, y ) = 1 and u D ( x, y ) = 2 log 3 .The right-hand side of the inequalities follows from Corollary 4.4 and Lemma 2.19.For the sharpness of the right-hand side of the inequalities, we consider the domain D = R n \ { e } . Let y = − x = − t e with 0 < t < u D ( x, y ) = 2 log 1 + 3 t √ − t and s D ( x, y ) = t . GROMOV HYPERBOLIC METRIC AND M ¨OBIUS TRANSFORMATIONS 11
Moreover, lim t → + u D ( x, y )log s D ( x,y )1 − s D ( x,y ) = lim t → + t √ − t log t − t = 3 . Thus completes the proof. (cid:3) the metric u D and M¨obius transformations In this section, we study quasi-invariance properties for the metric u D under M¨obiustransformations. We first give the distortion inequalities for the metric u D under M¨obiustransformations in arbitrary domains D ( R n . Theorem 6.1.
Let D and D ′ be proper subdomains of R n and f : R n → R n be a M¨obiustransformation with f D = D ′ . Then for x, y ∈ D , we have u D ( x, y ) ≤ u D ′ ( f ( x ) , f ( y )) ≤ u D ( x, y ) . (6.2) Proof.
The proof follows from the inequalities in Theorem 5.2 and M¨obius invariance ofthe metric δ D . (cid:3) Now we discuss the sharpness of inequalities (6.2) in some specific domains.
Theorem 6.3.
Let f be a M¨obius transformation with f B n = B n . Then for all x, y ∈ B n ,we have u B n ( x, y ) ≤ u B n ( f ( x ) , f ( y )) ≤ u B n ( x, y ) , and both inequalities are sharp.Proof. The double inequality is clear by Theorem 6.1.For the sharpness of the right-hand side of the inequalities, we consider f ( z ) = a ∗ + r ( z − a ∗ ) | z − a ∗ | , where a = te ( < t <
1) , a ∗ = a | a | , r = p | a ∗ | − x = − y = (1 − t ) e . Then f ( x ) = 2 t − − t + t e and f ( y ) = 11 + t − t e . Moreover, lim t → − u B n ( f ( x ) , f ( y )) u B n ( x, y ) = lim t → − log t − t + t √ t (2 − t )(1 − t + t )(1+ t − t ) log − tt = 3 . Thus the constant 3 is attained. The sharpness of the left-hand side of the inequalitiescan be seen by considering the inverse of f and hence the constant is also the bestpossible. (cid:3) Theorem 6.4.
Let f be a M¨obius transformation with f H n = B n . Then for all x, y ∈ H n ,we have u H n ( x, y ) ≤ u B n ( f ( x ) , f ( y )) ≤ u H n ( x, y ) , and the left-hand side of the inequalities is sharp. Proof.
The double inequality is clear by Theorem 6.1.For the sharpness of the left-hand side of the inequalities, we consider f ( z ) = − e n + 2( z + e n ) | z + e n | . Putting x = t e n and y = t e n with t > f ( x ) = − t − t + 1 e n and f ( y ) = t − t + 1 e n . Moreover, lim t → + u B n ( f ( x ) , f ( y )) u H n ( x, y ) = lim t → + t (cid:0) t − t (cid:1) = 13 . Thus completes the proof. (cid:3)
Next we give another type distortion inequalities for the metric u D under M¨obius trans-formations in some specific domains. Theorem 6.5.
Let f be a M¨obius transformation with f H n = H n . Then for all x, y ∈ H n ,we have u H n ( x, y ) − ≤ u H n ( f ( x ) , f ( y )) ≤ u H n ( x, y ) + 2 log 2 , and both inequalities are sharp.Proof. By Theorem 3.6 and Theorem 2.4, we obtain u H n ( f ( x ) , f ( y )) ≤ ρ H n ( f ( x ) , f ( y )) + 2 log 2 = ρ H n ( x, y ) + 2 log 2 ≤ u H n ( x, y ) + 2 log 2and u H n ( f ( x ) , f ( y )) ≥ ρ H n ( f ( x ) , f ( y )) = ρ H n ( x, y ) ≥ u H n ( x, y ) − . For the sharpness of the left-hand side of the inequalities, we consider f ( z ) = z | z | . Putting x = e + t e n and y = e + t e n with 0 < t < f ( x ) = 11 + t e + t t e n and f ( y ) = t t e + t t e n . Moreover,lim t → + ( u H n ( x, y ) − u H n ( f ( x ) , f ( y ))) = lim t → + (cid:18) (cid:18) t − t (cid:19) − (cid:18) − t t (cid:19)(cid:19) = 2 log 2 . Thus the left-hand side of the inequalities is sharp. The sharpness of the right-handside of the inequalities can be seen by considering the inverse of f . (cid:3) Theorem 6.6.
Let f be a M¨obius transformation with f H n = B n . Then for all x, y ∈ H n ,we have u H n ( x, y ) − ≤ u B n ( f ( x ) , f ( y )) ≤ u H n ( x, y ) + 2 log 2 , and the left-hand side of the inequalities is sharp. GROMOV HYPERBOLIC METRIC AND M ¨OBIUS TRANSFORMATIONS 13
Proof.
By Theorem 3.3, Theorem 3.6 and Theorem 2.4, we obtain u B n ( f ( x ) , f ( y )) ≤ ρ B n ( f ( x ) , f ( y )) + 2 log 2 = ρ H n ( x, y ) + 2 log 2 ≤ u H n ( x, y ) + 2 log 2and u B n ( f ( x ) , f ( y )) ≥ ρ B n ( f ( x ) , f ( y )) = ρ H n ( x, y ) ≥ u H n ( x, y ) − . For the sharpness of the left-hand side of the inequalities, we consider f ( z ) = − e n + 2( z + e n ) | z + e n | . Putting x = t e n and y = t e n with t > f ( x ) = − t − t + 1 e n and f ( y ) = t − t + 1 e n . Moreover,lim t →∞ ( u H n ( x, y ) − u B n ( f ( x ) , f ( y ))) = lim t →∞ (cid:18) (cid:18) t − t (cid:19) − t (cid:19) = 2 log 2 . Thus completes the proof. (cid:3)
Theorem 6.7.
Let f be a M¨obius transformation with f B n = B n . Then for all x, y ∈ B n ,we have u B n ( x, y ) − ≤ u B n ( f ( x ) , f ( y )) ≤ u B n ( x, y ) + 2 log 2 . Proof.
By Theorem 3.3 and Theorem 2.4, we obtain u B n ( f ( x ) , f ( y )) ≤ ρ B n ( f ( x ) , f ( y )) + 2 log 2 = ρ B n ( x, y ) + 2 log 2 ≤ u B n ( x, y ) + 2 log 2and u B n ( f ( x ) , f ( y )) ≥ ρ B n ( f ( x ) , f ( y )) = ρ B n ( x, y ) ≥ u B n ( x, y ) − . Thus completes the proof. (cid:3)
Acknowledgments.
This research was partly supported by National Natural ScienceFoundation of China (NNSFC) under Grant No.11771400 and No.11601485, and ScienceFoundation of Zhejiang Sci-Tech University (ZSTU) under Grant No.16062023 -Y.
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