## Abstract

Let (Formula presented.) be a domain that supports the (Formula presented.)-Poincaré inequality. Given a homeomorphism (Formula presented.), for (Formula presented.) we show that the domain (Formula presented.) 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 (Formula presented.) with non-empty boundary and for any conformal homeomorphism (Formula presented.) from the unit disc (Formula presented.) onto (Formula presented.) the complex derivative (Formula presented.) is integrable in the degree (Formula presented.). If (Formula presented.) is bounded then (Formula presented.). We prove that integrability in the degree (Formula presented.) is not possible for domains (Formula presented.) with infinite geodesic diameter.

Original language | English |
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Pages (from-to) | 247-256 |

Number of pages | 10 |

Journal | Computational Methods and Function Theory |

Volume | 14 |

Issue number | 2-3 |

DOIs | |

State | Published - 31 Oct 2014 |

## Keywords

- Brennan’s conjecture
- Sobolev homeomorphisms

## ASJC Scopus subject areas

- Analysis
- Computational Theory and Mathematics
- Applied Mathematics