Let $D\subset {ℂ}^n$ be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by $G_D(.,.)$. In this article we give for a compact subset $K\subset D$ a quantitative upper bound for the supremum ${sup}_{z\in K}\left|G_D(z,w)\right|$ in terms of the boundary distance of $K$ and $w$. This enables us to prove that, on a smooth bounded regular domain $D$ (in the sense of Diederich-Fornaess), the Bergman differential metric $B_D(w;X)$ tends to infinity, for $X\in {ℂ}^n/\left\{O\right\}$, when $w\in D$ tends to a boundary point....