Sobolev homeomorphisms and Brennan's conjecture

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a domain that supports the $p$-Poincar'e inequality. Given a homeomorphism $\phi \in L_p^1(\mathrm{\Omega })$, for $p>n$ we show the domain $\phi (\mathrm{\Omega })$ has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. {\bf The Inverse Brennan’s conjecture} states that for any simply connected plane domain ${\mathrm{\Omega }}^{\mathrm{\prime }}\subset \mathbb{C}$ with nonempty boundary and for any conformal homeomorphism $\phi$ from the unit disc $\mathbb{D}$ onto ${\mathrm{\Omega }}^{\mathrm{\prime }}$ the complex derivative ${\phi }^{\mathrm{\prime }}$ is integrable in the degree $s$, $-2<s<2\mathrm{/}3$. If ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is bounded than $-2<s\le 2$. We prove that integrability in the degree $s>2$ is not possible for domains ${\mathrm{\Omega }}^{\mathrm{\prime }}$ with infinite geodesic diameter.