Quasiconformal maps and substantial boundary points

Let D be a bounded domain in ℂ with ζ₀ ε ∂D. We say that ζ₀ is a substantial boundary point of D for the affine stretch x+iy → Kx+iy, where K > 1, if for every neighbourhood U of ζ₀ and for every component V of U ∩ D with ζ₀ ε ∂V, the maximal dilatation of f is at least K for every quasiconformal map f of V such that f(x + iy) = Kx + iy for all x + iy ε ∂V ∩ ∂D. We give here a criterion for a point ζ₀ to be a substantial boundary point for the affine stretch in D - Theorem 1.1 below. This will depend on the "narrowness" of D near ζ₀ though the particular way that D is narrow may vary, as we shall show.