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Through the Schwarz lemma, we provide a new point of view on three well-knownresults of the geometry of hyperbolic surfaces. The first result deal with thelength of closed geodesics on hyperbolic surfaces with boundary (Thurston,Parlier, Papadopoulos-Th\'eret). The two others give sharp lower bounds on twometric invariants: the length of the shortest non simple closed geodesic, andthe radius of the biggest embedded hyperbolic disk (Yamada). We also discuss aquestion of Papadopoulos and Th\'eret about the length of arcs on surfaces withboundary. In a sequel, we use a generalization of the Schwarz lemma due to Yauto study the injectivity radius of surfaces with bounded curvature.

Trois applications du lemme de Schwarz aux surfaces hyperboliques