## Abstract

Let Ω ⊂ R

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 Ω' ⊂ C with non-empty boundary and for any conformal homeomorphism ϕ from the unit disc D onto Ω' the complex derivative ϕ is integrable in the degree s, −2 <

^{n }be a domain that supports the p-Poincaré inequality. Given a homeomorphism ϕ ∈ L^{1}_{p}(Ω), for*p*> n we show that the domain ϕ() has finitegeodesic 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 Ω' ⊂ C with non-empty boundary and for any conformal homeomorphism ϕ from the unit disc D onto Ω' the complex derivative ϕ is integrable in the degree s, −2 <

*s*< 2/3. If Ω' is bounded then −2 < s ≤ 2. We prove that integrability in the degree*s*> 2 is not possible for domains Ω' with infinite geodesic diameter.Original language | English GB |
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Journal | Computational Methods and Function Theory |

State | Published - 1 Sep 2013 |

## Keywords

- Mathematics - Functional Analysis