Boundary behaviour of holomorphic functions on the cardioid domain with some applications

Abstract

The objective of this research paper is to show how the Bennan'sconjecture become a useful tool to construct a holomorphic function on the cardioid domain, and on the boundary of unit disk. Moreover , we have addressed some applications on the existence of cusp on the boundary of arising from integrability of conformalmaps through one of the polar function in the general solution of Laplace equation.