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We derive a priori second order estimates for fully nonlinear ellipticequations which depend on the gradients of solutions in critical ways onHermitian manifolds. The global estimates we obtained apply to an equationarising from a conjecture by Gauduchon which extends the Calabi conjecture;this was one of the original motivations to this work. We were also motivatedby the fact that there had been increasing interests in fully nonlinear pde'sfrom complex geometry in recent years, and aimed to develop general methods tocover as wide a class of equations as possible.

Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds