  We consider a homogeneous Bose-Fermi mixture, with the boson-fermion
interaction tuned by a Fano-Feshbach resonance, in the presence of mass and
density imbalance between the two species. By using many-body diagrammatic
methods, we first study the finite-temperature phase diagram for the specific
case of the mass-imbalanced mixture $^{87}$Rb $-^{40}$K for different values of
the density imbalance. We then analyse the quantum phase transition associated
with the disappearance at zero temperature of the boson condensate above a
critical boson-fermion coupling. We find a pronounced dependence of the
critical coupling on the mass ratio and a weak dependence on the density
imbalance. For a vanishingly small boson density, we derive, within our
approximation, the asymptotic expressions for the critical coupling in the
limits of small and large mass ratios. These expressions are relevant also for
the polaron-molecule transition in a Fermi mixture at small and large mass
ratios. The analysis of the momentum distribution functions at sufficiently
large density imbalances shows an interesting effect in the bosonic momentum
distribution due to the simultaneous presence of composite fermions and
unpaired fermions.
