A cosine inequality in the hyperbolic geometry

The main aim of this note is to show that the inequality hD2(x,y)≥hD2(x,z)+hD2(y,z)−2hD(x,z)hD(y,z)cos∠h(y,z,x) holds for any hyperbolic domain D⊂R2D⊂R2 and distinct points xx, y,z∈Dy,z∈D, where hDhD denotes the hyperbolic metric in DD and ∠h(y,z,x)∠h(y,z,x) the angle formed by the hyperbolic segments γh[z,x]γh[z,x] and γh[z,y]γh[z,y]. This shows that the answer to an open problem recently raised by Klén (2009) in [10] is positive.