We exhibit a class of bounded, strongly convex Hartogs domains with real-analytic boundary which are not Lu Qi-Keng, i.e. whose Bergman kernel function has a zero.

Let $D$ be a bounded strictly pseudoconvex domain in ${ℂ}^2$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$. This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in ${ℂ}^2$.