Integrability under an Inner Boundary Complex Domain

The purpose of this paper is generating a sharp condition on boundary ??C and w be a positive measurable function defined on ? which satisfy the assorted deterioration estimate at inner boundary of ? so the main result is Theorem (1), which comes as a corollary of the Koebe one quarter theorem and Koebe distortion theorem with discusses existence of positive constant K depending only on the modulus of z in ?, such that (1-|?(z) | )?K?(|z|) , for some z in ?, as well as , has been used this result as an essential tool to prove Theorems (2) and (3).