Let X be a complex space and φ: X → [-∞, ∞) an upper semicontinuous function on X. Consider the Hartogs domain Ω (X) given by Ω (X) = {(z, w) ∈ X × C: |w| < e-φ(z). In this article, some necessary and sufficient conditions on the complete hyperbolicity of Ω (X) are established.
