  It has been recently argued that the inclusive hadroproduction at the LHC of
a Higgs boson in association with a jet can be sensitive to the high-energy
dynamics. Moreover, the impact of the resummation at FCC energies is expected
to be large also in the inclusive cross section for the main Higgs production
channel in proton-proton collisions, namely the gluon fusion. As the energy
increases, a pure $k_t$-factorization framework or a collinear-factorization
approach supplemented by the high-energy resummation are adequate formalisms to
describe these processes. In both cases, the fundamental missing ingredient for
a next-to-leading logarithmic description is the Higgs boson impact factor. We
present the full next-to-leading result for the forward Higgs impact factor,
obtained in the infinite top-mass limit, discussing possible future extensions,
such as the calculation of the impact factor in the central rapidity region.
