  We study the Dirichlet problem of a class of fully nonlinear elliptic
equations on Hermitian manifolds and derive a priori $C^2$ estimates which
depend on the initial data on manifolds, the admissible subsolutions and the
upper bound of the gradients of the solutions. In some special cases, we obtain
the gradient estimates, and hence we can solve the corresponding Dirichlet
problem with admissible subsolutions. We also study the Hessian quotient
equations and $(m-1,m-1)$-Hessian quotient equations on compact Hermitian
manifolds without boundary.
