Abstract
The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern mathematical physics, the theory of partial differential equations, engineering, and electronics. In our present investigation, two subfamilies of starlike and bounded turning functions associated with a three-leaf-shaped domain were considered. These classes are denoted by BT3l and S-3l*, respectively. For the class BT3l, we study various coefficient type problems such as the first four initial coefficients, the Fekete-Szego and Zalcman type inequalities and the third-order Hankel determinant. Furthermore, the existing third-order Hankel determinant bounds for the second class will be improved here. All the results that we are going to prove are sharp.