A cosine inequality in the hyperbolic geometry (original) (raw)

Saminathan Ponnusamy

2010, Applied Mathematics Letters

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5 pages

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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) − 2h 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.