Dunlop's theorem

Dunlop's theorem provides conditions to determine the univalence (injectivity) of a certain type of complex function called a meromorphic function within a specific domain. Essentially, it states that if a function's behavior and certain associated functions meet particular criteria—such as the image of the domain under the function being convex or the function satisfying specific inequalities—then the function is univalent in that domain. This theorem is useful in complex analysis for establishing when functions are one-to-one, which is important for conformal mappings and understanding complex dynamics.