Negative holomorphic bisectional curvature of some bounded domains

We prove that a bounded domain in Cⁿ admitting a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature near the boundary, admits a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature everywhere. As a consequence we prove that strictly pseudoconvex bounded domains with C² boundary and bounded domains with squeezing function tending to 1 at every point of the boundary, admit a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature everywhere.