The strong Diederich-Fornæss index on C2 domains in Hermitian manifolds

For a relatively compact Stein domain Ω with $C^2$ boundary in a Hermitian manifold M, we consider the strong Diederich-Fornæss index, denoted $DF\left(\Omega \right)$: the supremum of all exponents $0<\eta <1$ such that eigenvalues of the complex Hessian of $-{(-\rho )}^{\eta }$ are bounded below by some positive multiple of ${(-\rho )}^{\eta }$ on Ω for some $C^2$ defining function ρ. We will show that $DF\left(\Omega \right)$ is completely characterized by the existence of a Hermitian metric with curvature terms satisfying a certain inequality when restricted to the null-space of the Levi-form.