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We study the injectivity radius of complete Riemannian surfaces (S,g) withcurvature |K(g)| bounded by 1. We show that if S is orientable with nonabelianfundamental group, then there is a point p in S with injectivity radius atleast arcsinh(2/\sqrt{3}). This lower bound is sharp independently of thetopology of S. This result was conjectured by Bavard who has already proved thegenus zero cases. We establish a similar inequality for surfaces with boundary.The proofs rely on a version due to Yau of the Schwarz lemma, and on the workof Bavard. This article is the sequel of a previous one where we studiedapplications of the Schwarz lemma to hyperbolic surfaces.

Le lemme de Schwarz et la borne sup\'erieure du rayon d'injectivit\'e des surfaces