On analytic families of conformal maps (2017)
Let Λ be a domain in C and let fλ(z) = z + a0(λ) + a1(λ)z −1 + ... be meromorphic in D∗ := {z ∈ C : |z| 1} ∪ {∞}. We assume that fλ(z) is holomorphic in λ ∈ Λ for fixed z.The main theorem states: Let Λ0 be a subdomain of Λ such that fλ is univalent in D∗ for λ ∈ Λ0. If fλ0 has a quasiconformal extension to the closure of D∗ for one λ0 ∈ Λ0 then fλ has a quasiconformal extension for all λ ∈ Λ0.This result is related to a theorem of Mañé, Sad and Sullivan (1983) where the assumptions are however different. The main tool of our proof is the Grunsky inequality for univalent functions.