A cosine inequality in the hyperbolic geometry

The main aim of this note is to show that the inequality h D 2 ( x , y ) ≥ h D 2 ( x , z ) + h D 2 ( y , z ) − 2 h D ( x , z ) h D ( y , z ) cos ∠ h ( y , z , x ) holds for any hyperbolic domain D ⊂ R 2 and distinct points x , y , z ∈ D , where h D denotes the hyperbolic metric in D and ∠ h ( y , z , x ) the angle formed by the hyperbolic segments γ h [ z , x ] and γ h [ z , y ] . This shows that the answer to an open problem recently raised by Klén (2009) in [10] is positive.