  We examine cosmological inflation in a broad family of scalar-tensor models
characterized by scalar-dependent non minimal kinetic couplings and
Gauss-Bonnet terms. Using a slow roll-approximation, we compute in detail
theoretical expectations of observables as spectral indexes, scalar-to-tensor
ratio, their running and their running of the running in terms of the
parameters which characterize the scalar-tensor model. Hierarchies of
consistency equations relating scalar and tensor pertubations and higher order
running parameters are presented and examined at the slow roll approximation
for the kind of models of interest in this work. From We find detailed
expressions for constraints among these parameters. For a specific model, we
analyse such quantities and make contact with latest Planck observational data .
