  An approximate expression for the inclusive Higgs production cross section in
gluon fusion at NNNLO in QCD with finite top mass is presented. We argue that
an accurate approximation can be constructed combining (and improving) the
large- and small-z behaviours of the partonic cross section, which are both
known to all orders from soft-gluon (Sudakov) and high-energy (BFKL)
resummations, respectively. For a 125 GeV Higgs at LHC at 8 TeV, we find an
increase of about 6-13% with respect to the NNLO inclusive cross section for
the conventional scale \mu_R=m_H/2, suggesting that higher order QCD
corrections might be underestimated by presently available results. We also
find a significant reduction of the scale uncertainty.
