Bogomolov inequality for Higgs parabolic bundles

We show that over some smooth projective varieties every semistable Higgs logarithmic vector bundle is semistable in the ordinary sense, hence satisfies Bogomolov inequality. More generaly, we prove that semistable Higgs parabolic vector bundles of rank two over smooth projective varieties of dimension ≥ 2 satisfy the "parabolic" 'Bogomolov inequality.