Univalence criteria and quasiconformal extensions

Let f be a locally univalent meromorphic function in the unit disk Δ. Recently, Epstein obtained a differential geometric proof for the fact that if f satisfies an inequality involving a suitable real-valued function σ, then f is univalent in Δ and has a quasiconformal extension to the sphere. We give a more classical proof for this result by means of an explicit quasiconformal extension, and obtain generalizations of the result under suitable conditions even if σ is allowed to be complex-valued and Δ is replaced by a quasidisk.