Hartogs Domains and the Diederich Forn{\ae}ss Index

We study a geometric property of the boundary on Hartogs domains which can be used to find upper and lower bounds for the Diederich-Forn{\ae}ss Index. Using this, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich-Forn{\ae}ss Index for a Hartogs domain is equal to one if and only if the domain admits a family of good vector fields in the sense of Boas and Straube. We also study the analogous problem for a Stein neighborhood basis, and show that under the same hypotheses if the Diederich-Forn{\ae}ss Index for a Hartogs domain is equal to one then the domain admits a Stein neighborhood basis.