A proof of Sethares’ conjecture

Let ϕ(z) be holomorphic in the unit disk Δ and meromorphic on Δ. Suppose f is a Teichmüller mapping with complex dilatation kϕ/|ϕ|. In 1968, Sethares conjectured that f is extremal if and only if either (i) ϕ has a double pole or (ii) ϕ has no pole of order exceeding two on ∂ϕ. The “if” part of the conjecture had been solved by himself. We will give the affirmative answer to the “only if” part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the “if” part of Sethares’ conjecture and improves the result by Reich and Shapiro in 1990.