A Kaup-Upmeier type theorem for possibly unbounded non-hyperbolic Reinhardt domains via Bergman theoretic inequalities

A theorem due to Kaup and Upmeier states that two bounded balanced pseudoconvex domains are biholomorphic if and only if they are linearly equivalent. In this article, we prove that this theorem can occur for possibly unbounded non-hyperbolic Reinhardt domains under certain Bergman theoretic conditions. We also find explicit unbounded non-hyperbolic examples of our theorem.