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  Applied Mathematics Letters 23 (2010) 887–891 Contents lists available at ScienceDirect Applied Mathematics Letters  journal homepage: www.elsevier.com/locate/aml A cosine inequality in the hyperbolic geometry  M. Huang a , S. Ponnusamy b , H. Wang a , X. Wang a, ∗ a Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China b Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India a r t i c l e i n f o  Article history: Received 24 May 2009Received in revised form 29 March 2010Accepted 5 April 2010 Keywords: Hyperbolic metricQuasihyperbolic metricHyperbolic geodesicQuasihyperbolic geodesicA cosine inequality a b s t r a c t The main aim of this note is to show that the inequality  h 2 D (  x ,  y )  ≥  h 2 D (  x ,  z  )  +  h 2 D (  y ,  z  )  − 2 h D (  x ,  z  ) h D (  y ,  z  ) cos     h (  y ,  z  ,  x ) holdsforanyhyperbolicdomain D  ⊂  R 2 anddistinctpoints  x ,  y ,  z   ∈  D , where  h D  denotes the hyperbolic metric in  D  and      h (  y ,  z  ,  x )  the angle formedby the hyperbolic segments  γ  h [  z  ,  x ]  and  γ  h [  z  ,  y ] . This shows that the answer to an openproblem recently raised by Klén (2009) in [10] is positive. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction and main results Throughout this work we assume that  D  is a domain in the  n -dimensional Euclidean space R n ( n  ≥  2) and B n (  x 0 , r  )  ={  x  ∈  R n : |  x  −  x 0 |  <  r  }  is the open ball at  x 0  of radius  r   >  0. For convenience, we denote B n ( 0 , 1 )  by B n and, in particular, B 2 by B . We call  D  a  hyperbolic domain  if Card ( R n \  D )  ≥  2. In the following, we identify R 2 with C .The  hyperbolic   (or  Poincaré )  density  on  D  ⊂  C is given by ρ D (  z  )  =  ρ B (  g  (  z  )) |  g   (  z  ) | , where  ρ B (  z  )  =  2 /( 1  − |  z  | 2 )  and  g  : D  →  B is conformal. Then the differential  ρ D (  z  ) | d  z  |  defines the hyperbolic metric of  D  and it is well-known that  ρ D  is a conformal invariant, for it does not depend on the particular choice of the mapping  g  ;cf. [1, Section 4.6] and [2]. Foranypairofpoints  z  1 ,  z  2  in D  ⊂  C andpath α  ⊂  D from  z  1  to  z  2 ,wedefinethe hyperboliclengthh D (α) andthe hyperbolic distance h D (  z  1 ,  z  2 )  from  z  1  to  z  2  as follows: h D (α)  =   α ρ D (  z  ) | d  z  |  and  h D (  z  1 ,  z  2 )  =  inf  α h D (α), where the infimum is taken over all rectifiable curves  α  in  D  joining  z  1  to  z  2 . The minimal hyperbolic length path is calledthe  hyperbolic geodesic  .Conformal invariants and conformally invariant metrics play important roles in geometric function theory. One of themost important conformally invariant metrics is the hyperbolic metric of the unit ball  B n or the half-space  H n (detailedstudies of hyperbolic metrics and many related topics may be found in the books of Vuorinen [3] and Beardon [4]).  The research was partly supported by NSF of China (No. 10771059) and Tianyuan Foundation (No. 10926068). ∗  Corresponding author. E-mail addresses:  mzhuang79@yahoo.com.cn (M. Huang), samy@iitm.ac.in (S. Ponnusamy), wanghui.19850824@yahoo.com.cn (H. Wang), xtwang@hunnu.edu.cn (X. Wang).0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.aml.2010.04.004  888  M. Huang et al. / Applied Mathematics Letters 23 (2010) 887–891 In the planar case, the hyperbolic metric can be defined for all hyperbolic domains. The same method of definition doesnot work in the higher dimension and therefore one might search for alternate approaches.The  quasihyperbolic length  of a rectifiable path  γ   in  D   R n is the number (cf. [5,6])  k (γ)  =   γ  | d  z  | d (  z  ), where  d (  z  )  :=  dist (  z  ,∂ D ) , the distance from  z   to the boundary  ∂ D  of   D . The  quasihyperbolic distance k D (  z  1 ,  z  2 )  from  z  1  ∈  D to  z  2  ∈  D  is defined by k D (  z  1 ,  z  2 )  =  inf   k (γ), in which the infimum is taken over all arcs  γ   joining  z  1  to  z  2  in  D .Gehring and Palka [5] introduced the quasihyperbolic metric in a domain  D   R n . Many of the basic properties of thismetric may be found in [7,5,8,6]. Recall that an arc  γ   from  z  1  to  z  2  is a  quasihyperbolic geodesic   if    k (γ)  =  k D (  z  1 ,  z  2 ) . Eachsubarc of a quasihyperbolic geodesic is obviously a quasihyperbolic geodesic. It is known that a quasihyperbolic geodesicbetween any two points in R n exists; see [7, Lemma 1]. For any  x ,  y  ∈  D , let  γ  h [  x ,  y ]  (resp.  γ  k [  x ,  y ] ) denote the hyperbolic (resp. quasihyperbolic) geodesic segment in  D  withthe endpoints  x  and  y . For any  x ,  y ,  z   ∈  D ,      h (  x ,  y ,  z  )( ∈ [ 0 ,π ] )  (resp.      k (  x ,  y ,  z  ) ) denotes the angle formed by  γ  h [  y ,  x ]  and γ  h [  y ,  z  ]  (resp.  γ  k [  y ,  x ]  and  γ  k [  y ,  z  ] ). Definition 1.  A hyperbolic (resp. quasihyperbolic) geodesic trigon  T   with vertices  x ,  y  and  z   is  γ  h [  x ,  y ] ∪  γ  h [  y ,  z  ] ∪  γ  h [  z  ,  x ] (resp.  γ  k [  x ,  y ]∪ γ  k [  y ,  z  ]∪ γ  k [  z  ,  x ] ). The interior of a hyperbolic (resp. quasihyperbolic) geodesic trigon is the set of points in D  that is enclosed by the hyperbolic (resp. quasihyperbolic) geodesic trigon.If the interior of the hyperbolic (resp. quasihyperbolic) geodesic trigon is simply connected we call  T   a hyperbolic(resp. quasihyperbolic) triangle, which is denoted by  h (  x ,  y ,  z  )  (resp.  k (  x ,  y ,  z  ) ). Otherwise  T   is called a hyperbolic (resp.quasihyperbolic) trigon, denoted by   ∗ h (  x ,  y ,  z  )  (resp.   ∗ k (  x ,  y ,  z  ) ).It is well-known that the Law of Cosines is a fundamental tool in Euclidean geometry (cf. [9, Chapter 10]). In [4], on the basis of the hyperbolic functions, Beardon considered the Law of Cosines in the hyperbolic geometry (see [4, Section 7.12]) Klén also considered the Law of Cosines in the hyperbolic geometry and quasihyperbolic geometry. In particular, thefollowing results are obtained, which are not related to hyperbolic functions.  Theorem A   ( [10, Theorem 4.1] ) .  Let D  =  R 2 \ { 0 }  and x ,  y ,  z   ∈  D. Then: (i)  for the quasihyperbolic triangle   k (  x ,  y ,  z  ) ,k 2 D (  x ,  y )  =  k 2 D (  x ,  z  )  +  k 2 D (  y ,  z  )  −  2 k D (  x ,  z  ) k D (  y ,  z  ) cos     k (  y ,  z  ,  x ) ;  (1)(ii)  for the quasihyperbolic trigon   ∗ k (  x ,  y ,  z  ) ,k 2 D (  x ,  y )  =  k 2 D (  x ,  z  )  +  k 2 D (  y ,  z  )  −  2 k D (  x ,  z  ) k D (  y ,  z  ) cos     k (  y ,  z  ,  x )  −  4 π(π  −  α), where  α  =      (  x , 0 ,  y ) .  Theorem B  ( [10, Theorem 4.27] ) .  Let x ,  y ,  z be three distinct points in H 2 . Thenk 2 H 2 (  x ,  y )  ≥  k 2 H 2 (  x ,  z  )  +  k 2 H 2 (  y ,  z  )  −  2 k H 2 (  x ,  z  ) k H 2 (  y ,  z  ) cos     k (  y ,  z  ,  x ).  (2)The following example shows that the identity (1) in Theorem A does not hold in the general domain. Example 1.  Let  D  =  R 2 \ {− 1 , 1 } , and let  z  1  = − 12 ,  z  2  = − 32 ,  z  3  = − 1  −  12 i ,  z  4  =  0 and  z  5  =  i . Then:(a)  k 2 D (  z  1 ,  z  3 ) <  k 2 D (  z  1 ,  z  2 )  +  k 2 D (  z  2 ,  z  3 )  −  2 k D (  z  1 ,  z  2 ) k D (  z  2 ,  z  3 ) cos     k (  z  1 ,  z  2 ,  z  3 ) ;(b)  k 2 D (  z  1 ,  z  2 )  =  k 2 D (  z  1 ,  z  3 )  +  k 2 D (  z  2 ,  z  3 )  −  2 k D (  z  1 ,  z  3 ) k D (  z  2 ,  z  3 ) cos     k (  z  1 ,  z  2 ,  z  3 ) ;(c)  k 2 D (  z  1 ,  z  5 ) >  k 2 D (  z  4 ,  z  1 )  +  k 2 D (  z  4 ,  z  5 )  −  2 k D (  z  4 ,  z  1 ) k D (  z  4 ,  z  5 ) cos     k (  z  1 ,  z  4 ,  z  5 ) . Solution.  The proofs of (a) and (b) follow from similar arguments in [10] and the facts that k D (  z  1 ,  z  2 )  =  π,  k D (  z  2 ,  z  3 )  =  k D (  z  1 ,  z  3 )  = π 2and     k (  z  1 ,  z  3 ,  z  2 )  =      k (  z  1 ,  z  2 ,  z  3 )  =  π. Further, as k D (  z  4 ,  z  1 )  =  log2 ,  k D (  z  1 ,  z  5 )  =   (π/ 4 ) 2 +  log 2  √  2 ,  k D (  z  4 ,  z  5 )  =  log 1  +√  22 ,  M. Huang et al. / Applied Mathematics Letters 23 (2010) 887–891  889 and     k (  z  1 ,  z  4 ,  z  5 )  = π 2 , the proof of (c) follows.It is natural to ask whether Theorem B holds in the hyperbolic geometry. This is indeed proposed as an open problem byKlén [10, Open problem 4.64] in the following form. Question 1.  Does the inequality h 2 D (  x ,  y )  ≥  h 2 D (  x ,  z  )  +  h 2 D (  y ,  z  )  −  2 h D (  x ,  z  ) h D (  y ,  z  ) cos     h (  y ,  z  ,  x )  (3)hold for all simply connected hyperbolic domains  D  ⊂  R n and any distinct points  x ,  y ,  z   ∈  D ?Themainaimofthisworkistopresentaproofof (3).InviewofLiouville’stheorem,itsufficestoconsiderthecase n  =  2in Question 1.  Theorem 1.  Let D  ⊂  R 2 be a simply connected hyperbolic domain. Then for any three distinct points x ,  y ,  z   ∈  D, theinequality  (3)  holds. For the proof of  Theorem 1, we need some preparation. Lemma 1.  For any distinct points z  1 ,  z  2 ,  z  3  ∈  B , we haveh 2 B (  z  1 ,  z  2 )  ≥  h 2 B (  z  1 ,  z  3 )  +  h 2 B (  z  2 ,  z  3 )  −  2 h B (  z  1 ,  z  3 ) h B (  z  2 ,  z  3 ) cos     h (  z  2 ,  z  3 ,  z  1 ).  (4) Proof.  Since  h B  is a conformal invariant, without loss of generality, we may assume that  z  3  =  0 and  z  1  =  r  1  >  0. Now, welet  z  2  =  r  2 e i θ  , where  θ   ∈ [ 0 ,π ] . Then | 1  −  z  2  z  1 | =   r  21 r  22  +  1  −  2 r  1 r  2  cos θ, |  z  2  −  z  1 | =   r  21  +  r  22  −  2 r  1 r  2  cos θ  and h B (  z  1 ,  z  2 )  =  log  | 1  −  z  2  z  1 | + |  z  2  −  z  1 || 1  −  z  2  z  1 | − |  z  2  −  z  1 |=  log   r  21 r  22  +  1  −  2 r  1 r  2  cos θ   +   r  21  +  r  22  −  2 r  1 r  2  cos θ   2 ( 1  −  r  21 )( 1  −  r  22 ). Also we see that h B (  z  1 , 0 )  = 12log 1  +  r  1 1  −  r  1 and  h B (  z  2 , 0 )  = 12log 1  +  r  2 1  −  r  2 . To prove (4), it is sufficient to prove the inequality log 2   r  21 r  22  +  1  −  2 r  1 r  2  cos θ   +   r  21  +  r  22  −  2 r  1 r  2  cos θ   2 ( 1  −  r  21 )( 1  −  r  22 ) ≥  log 2  1  +  r  1 1  −  r  1 +  log 2  1  +  r  2 1  −  r  2 −  2log 1  +  r  1 1  −  r  1 log 1  +  r  2 1  −  r  2 cos θ.  (5)To prove (5), we let  x  =  cos θ   and consider the function  f  (  x )  =  log 2   r  21 r  22  +  1  −  2 r  1 r  2  x  +   r  21  +  r  22  −  2 r  1 r  2  x  2 ( 1  −  r  21 )( 1  −  r  22 ) −  log 2  1  +  r  1 1  −  r  1 −  log 2  1  +  r  2 1  −  r  2 +  2  x log 1  +  r  1 1  −  r  1 log 1  +  r  2 1  −  r  2 .  890  M. Huang et al. / Applied Mathematics Letters 23 (2010) 887–891 By a straightforward computation we have that  f   (  x )  = − 4 r  1 r  2  log (   r  21 r  22  +  1  −  2 r  1 r  2  x  +   r  21  +  r  22  −  2 r  1 r  2  x ) 2 ( 1  −  r  21 )( 1  −  r  22 )  × 1   r  21 r  22  +  1  −  2 r  1 r  2  x   r  21  +  r  22  −  2 r  1 r  2  x +  2log 1  +  r  1 1  −  r  1 log 1  +  r  2 1  −  r  2 and  f   (  x )  = 4 r  21 r  22 ( r  21 r  22  +  1  −  2 r  1 r  2  x )( r  21  +  r  22  −  2 r  1 r  2  x ) ×  2  −   r  21 r  22  +  1  −  2 r  1 r  2  x   r  21  +  r  22  −  2 r  1 r  2  x +   r  21  +  r  22  −  2 r  1 r  2  x   r  21 r  22  +  1  −  2 r  1 r  2  x  ×  log   r  21 r  22  +  1  −  2 r  1 r  2  x  +   r  21  +  r  22  −  2 r  1 r  2  x  2 ( 1  −  r  21 )( 1  −  r  22 )  . For  y  ∈ [ 0 , 1 ) , we introduce  g  (  y )  =  log 1  +  y 1  −  y − 2  y 1  +  y 2 . Itcanbeeasilyseenthat  g   isincreasingon ( 0 , 1 ) .Inparticular,  g  (  y )  ≥  g  ( 0 )  =  0.Thisobservationshowsthatfor a  >  b  >  0,we have12log  a  +  ba  −  b ≥ aba 2 +  b 2 and therefore, this inequality holds if  a  =   r  21 r  22  +  1  −  2 r  1 r  2  x  and  b  =   r  21  +  r  22  −  2 r  1 r  2  x . The last inequality implies that  f   (  x ) <  0 for  x  ∈  ( − 1 , 1 ) . Moreover, as  f  ( − 1 )  =  f  ( 1 )  =  0, Rolle’s mean value theoremshows that the function  f   is non-negative on  [− 1 , 1 ]  and hence, (5) follows. This completes the proof.   Example 2.  We note that strict inequality can occur in Lemma 1. In order to see this, we let  z  1  =  12  and  z  2  =  i2 . Then h 2 B (  z  1 ,  z  2 ) >  h 2 B (  z  1 , 0 )  +  h 2 B (  z  2 , 0 )  −  2 h B (  z  1 , 0 ) h B (  z  2 , 0 ) cos     h (  z  1 , 0 ,  z  2 ).  (6)The proof of inequality (6) easily follows from the following equalities: h B (  z  1 ,  z  2 )  =  log 25  +  4 √  349 ,  h B (  z  1 , 0 )  =  h B (  z  2 , 0 )  =  log3and θ   =      h (  x , 0 ,  y )  = π 2 . Proof of Theorem 1.  Since h D  isaconformalinvariant,theproofof Theorem1followsfromtheRiemannmappingtheoremand Lemma 1.   References [1] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, 1992.[2] L. Keen, N. Lakic, Hyperbolic geometry from a local viewpoint, in: London Mathematical Society Student Texts, vol.. 68, Cambridge university press,Cambridge, 2007.[3] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, in: Lecture Notes in Math., vol. 1319, Springer-Verlag, Berlin, 1988.

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