For s not zero, the Bergman metric on these Hartogs domains is Kähler-Einstein if and only if the domain is biholomorphic to a ball and the base is a ball with s=1/(n+1).
Symplectic geometry of Cartan-Hartogs domains
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Bergman--Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains
For s not zero, the Bergman metric on these Hartogs domains is Kähler-Einstein if and only if the domain is biholomorphic to a ball and the base is a ball with s=1/(n+1).