  We consider the combined effect of a gap and a Zeeman interaction on the
helical Dirac fermions which exist on the surface of a topological insulator.
Magneto-optical properties, the magnetization, Hall effect and the density of
states are considered with emphasis on the particle-hole asymmetry which arises
when a subdominant Schr\"odinger piece is included along with the dominant
Dirac part of the Hamiltonian. When appropriate, we compare our results with
those of a single valley gapped graphene system for which Zeeman splitting
behaves differently. We provide a derivation of the phase offset in the
magnetic oscillations brought about by the combined effect of the gap and
Schr\"odinger term without requiring the semiclassical Onsager quantization
condition. Our results agree with previous discussions based on semiclassical
arguments.

Mesoscale and Nanoscale Physics