  We are concerned with fully nonlinear elliptic equations on complex manifolds
and search for technical tools to overcome difficulties in deriving a priori
estimates which arise due to the nontrivial torsion and curvature, as well as
the general (non-pseudoconvex) shape of the boundary. We present our methods,
which work for more general equations, by considering a specific equation which
resembles the complex Monge-Ampere equation in many ways but with crucial
differences. Our work is motivated by recent increasing interests in fully
nonlinear equations on complex manifolds from geometric problems.
