Negative holomorphic bisectional curvature of some bounded domains

We prove that a bounded domain in ${\mathbb{C}}^n$ admitting a complete Kähler metric with negatively pinched holomorphic bisectional curvature near the boundary, admits a complete Kähler metric with negatively pinched holomorphic bisectional curvature everywhere. As a consequence we prove that strictly pseudoconvex bounded domains with $C^2$ boundary and bounded domains with squeezing function tending to 1 at every point of the boundary, admit a complete Kähler metric with negatively pinched holomorphic bisectional curvature everywhere.