Sobolev Homeomorphisms and Brennan's Conjecture

Let be a domain that supports the -Poincar, inequality. Given a homeomorphism , for we show that the domain has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. The Inverse Brennan's conjecture states that for any simply connected plane domain with non-empty boundary and for any conformal homeomorphism from the unit disc onto the complex derivative is integrable in the degree . If is bounded then . We prove that integrability in the degree is not possible for domains with infinite geodesic diameter.