Comparison of the Bergman and Szegő kernels on arbitrary bounded pseudoconvex domains with $C^2$ boundary

On every bounded pseudoconvex domain $\mathrm{\Omega }⋐{\mathbb{C}}^n$, $n\ge 2$, with $C^2$ boundary, we prove $K_{\mathrm{\Omega }}\mathrm{/}{S}_{\mathrm{\Omega }}\le C_{\mathrm{\Omega },\epsilon }{\delta }_{\mathrm{\Omega }}^{-1}\mathrm{\mid }\log {\delta }_{\mathrm{\Omega }}{\mathrm{\mid }}^{1+\epsilon }$ near the boundary for every $\epsilon >0$, where ${\delta }_{\mathrm{\Omega }}$ denotes the boundary distance. For each positive nondecreasing $H(s)=o(s)$, a smoothly bounded pseudoconvex complete Reinhardt domain in ${\mathbb{C}}^2$ shows that the logarithmic factor cannot be replaced by $H(\mathrm{\mid }\log {\delta }_{\mathrm{\Omega }}\mathrm{\mid })$.