Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in ${\mathbb C}^n$ is bounded from above by $δ|\logδ|^p$ for any $p>n$, where $δ$ 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 $δ|\logδ|^p$ for any $p<-1$. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of $δ$.

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