  We investigate the mixing-demixing transition and the collapse in a
quasi-two-dimensional degenerate boson-fermion mixture (DBFM) with a bosonic
vortex. We solve numerically a quantum-hydrodynamic model based on a new
density functional which accurately takes into account the dimensional
crossover. It is demonstrated that with the increase of interspecies repulsion,
a mixed state of DBFM could turn into a demixed state. The system collapses for
interspecies attraction above a critical value which depends on the vortex
quantum number. For interspecies attraction just below this critical limit
there is almost complete mixing of boson and fermion components. Such mixed and
demixed states of a DBFM could be experimentally realized by varying an
external magnetic field near a boson-fermion Feshbach resonance, which will
result in a continuous variation of interspecies interaction.
