The Strong Diederich-Fornæss Index on $C^2$ Domains in Hermitian Manifolds

For a relatively compact Stein domain $\mathrm{\Omega }$ with $C^2$ boundary in a Hermitian manifold $M$, we consider the strong Diederich-Fornæss index, denoted $DF(\mathrm{\Omega })$: 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 $\mathrm{\Omega }$ for some $C^2$ defining function $\rho$. We will show that $DF(\mathrm{\Omega })$ 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.