We consider boundedness of a function defined by an infinite product which is used to study a uniqueness theorem on a plane domain and the point separation problem of a two-sheeted covering Riemann surface. We show that there is such an infinite product that it converges but the function defined by it is not bounded on arbitrary Zalcman domain.
Yasuyuki Kobayashi. "On boundedness of a function on a Zalcman domain." Proc. Japan Acad. Ser. A Math. Sci. 77 (1) 22 - 24, Jan. 2001. https://doi.org/10.3792/pjaa.77.22