  We in this paper investigate the phase diagram associated with the BCS-BEC
crossover of a three-component ultracold superfluid-Fermi-gas of different
chemical-potentials and equal masses in two dimensions. The gap order parameter
and number densities are found analytically by using the functional
path-integral method. The balance of paring will be broken in the free space
due to the unequal chemical-potentials. We obtain the same particle
number-density and condensed fraction in the BCS superfluid phase as that in a
recent paper (Phys. Rev. A 83, 033630), while the Sarma phase of coexistence of
normal and superfluid Fermi gases is the characteristics of inhomogeneous
system. The minimum ratio of BCS superfluid phase becomes 1/3 in the BCS limit
corresponding to the zero-ratio in the two-component system in which the
critical point of phase separation is {\epsilon}B/{\epsilon}F = 2 but becomes 3
in the three-component case.
