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Dragutin Svrtan and Darko Veljan, Non-Euclidean versions of some classical triangle inequalities,
Forum Geometricorum, 12 (2012) 197--209.
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Abstract. In this paper we recall with short proofs of some classical triangle inequalities, and prove corresponding non-Euclidean, i.e., spherical and hyperbolic versions of these inequalities. Among them are the well known Euler's inequality, Rouché's inequality (also called ``the fundamental triangle inequality"), Finsler - Hadwiger's inequality, isoperimetric inequality and others.

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