Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes

We consider Roumieu-Carleman ultraholomorphic classes and classes of functions admitting asymptotic expansion in unbounded sectors, defined in terms of a log-convex sequence M. Departing from previous results by S. Mandelbrojt and B. Rodríguez-Salinas, we completely characterize the injectivity of the Borel map by means of the theory of proximate orders: A growth index ω(M) turns out to put apart the values of the opening of the sector for which injectivity holds or not. In the case of surjectivity, we considerably extend partial results by J. Schmets and M. Valdivia and by V. Thilliez, and prove a similar dividing character for the index γ(M) (introduced by Thilliez, and generally different from ω(M)) in some standard situations (for example, as far as M is strongly regular).