Comparison of the Bergman and Szeg\"{o} kernels

The quotient of the Szeg"{o} and Bergman kernels for a smooth bounded pseudoconvex domains in ${\mathbb{C}}^n$ is bounded from above by $\delta \mathrm{\mid }\log \delta {\mathrm{\mid }}^p$ for any $p>n$, where $\delta$ is the distance to the boundary. For a class of domains that includes those of D’Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by $\delta \mathrm{\mid }\log \delta {\mathrm{\mid }}^p$ for any $p<-1$. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of $\delta$.