Jordan domains with a rectifiable arc in their boundary

We show that if an open arc J of the boundary of a Jordan domain $\mathrm{\Omega }$ is rectifiable, then the derivative $\mathrm{\Phi }$’ of the Riemann map $\Phi: D\rightarrow \Omega$fromtheopenunitdiskDonto$\Omega$behavesasan$H^1$ function when we approach the arc ${\mathrm{\Phi }}^{-1}(J^{\mathrm{\prime }})$,where $J^{\mathrm{\prime }}$ is any compact subarc of $J$. "