Some Results on Koebe one- quarter theorem and Koebe distortion theorem

Abstract

n this research paper, we consider that , be a conformal mapping from a simply connected domain onto unit disk

with a great role for each of Koebe one- quarter theorem and Koebe distortion theorem to prove there is a constant depending on

the modulus of in such that for some

In addition, this result generates a sharp result on the integrability of gradient of Cauchy transform over a sequence which

is getting larger in , such that