  We analytically study the ground-state phase diagrams of ultracold bosons
with various values of the effective magnetic quantum number $m$ in a
state-dependent hexagonal optical lattice by using the generalized
effective-potential Landau theory, where the site-offset energy between the two
triangular sublattice A and B is tunable. Our analytical calculations of
third-order corrections are in reasonably good agreement with the previous
cluster Gutzwiller calculations. Furthermore, we reveal the reason why the
regions of the Mott lobes $(n,n)$ in phase diagrams for $m=0.02$ are
unexpectedly expanded with increasing $J/U$ in deep lattice.
