how to determine the pseudoconvexity of Hartog domains

let $\Omega\subset C^n$ be a domain with $C^\infty$ defining function $-\phi$ and $-\psi$ ,then we consider the set $\bar{\Omega}=(z_1,z_2,z_3)\in C^{n+d_2+d_3}:{|z_2|^2\over \phi(z_1)}+{|z_3|^2\over \psi(z_1) }<1$ , then what is the sufficient and necessary conditions for $\bar{\Omega}$ being pseudoconvex and why ?

An a special type of domain in complex Euclidean space , the questions as above have their own meaning , so does there any one studied this type question systematically and any reference ?

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