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

Abstract

In 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