Abstract. We consider Carleman-Roumieu ultraholomorphic classes in sectors of the Riemann surface of the logarithm, consisting of holomorphic functions with global bounds for their derivatives in terms of a sequence ${\bf M}=(M_n)_{n=0}^{\infty}$ ($M_n>0$). This control on the derivatives is deeply linked with the existence of an asymptotic expansion for such functions at the vertex of the sector. The study of the injectivity and surjectivity of the Borel map, sending every such function to the sequence of its derivatives at the vertex, is not complete in this context, unlike the ultradifferentiable one with the classical results of Denjoy-Carleman and Petzsche. We will highlight the importance of several indices of O-regular variation of ${\bf M}$ in this problem. For sequences ${\bf M}$ of moderate growth, or of rapid growth (in a sense to be made precise), these indices determine the opening of the sector above which, respectively below which, the Borel map is injective, resp. surjective.

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