Locally biholomorphic mappings of multiconnected domains

Sufficient conditions were found in [1, 2] for a domain in the complex plane to admit a finitely valent locally biholomorphic mapping of the whole plane. In this article we find the relevant necessary and sufficient conditions. Also, we answer the question that was posed by Aksent'ev and Ul'yanov in connection with the problem under consideration. The answer yields a lower bound for the valency while generalizing a result of [3] to polydisks.