The Neumann problem for a class of mixed complex Hessian equations

In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations \begin{document}$ \sigma_k(\partial \bar{\partial} u) = \sum\limits _{l = 0}^{k-1} \alpha_l(z) \sigma_l (\partial \bar{\partial} u) $\end{document} with \begin{document}$ 2 \leq k \leq n $\end{document} , and establish the global \begin{document}$ C^1 $\end{document} estimates and reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global \begin{document}$ C^2 $\end{document} estimates and the existence theorems when \begin{document}$ k = n $\end{document} .