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In this paper, we prove the following result : let X be a complex manifold,hyperbolic for the Carath\'eodory distance and let U be an open set relativelycompact in X. Then, there exists k<1 such that we get, for the Carath\'eodoryinfinitesimal metric E_X(x,v) less or equal to kE_U(x,v). We also get resultsconcerning fixed points of holomorphic mappings from X to U.

Distances invariantes et points fixes d'applications holomorphes