One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions $\mathcal{X}$ defined in the new domain, including the sharp estimates for the coefficients $a_2,a_3$, and $a_4$, and for three second-order and third-order Hankel determinants, ${\mathcal{H}}_{2,1}\mathcal{X},{\mathcal{H}}_{2,2}\mathcal{X}$, and ${\mathcal{H}}_{3,1}\mathcal{X}$. The optimality of each obtained estimate is given as well.

In the current article, we utilize the concept of subordination to establish a new subclass of analytic functions associated with a bounded domain that is symmetric about the real axis. By applying the convolution technique, we derive the necessary and sufficient condition, the radius of convexity for this recently introduced class. Furthermore, we prove the sharp upper bounds for the second-order Hankel determinants $|{\mathcal{H}}_{2,1}\xi |,|{\mathcal{H}}_{2,2}\xi |$ and third-order Hankel determinant $|{\mathcal{H}}_{3,1}\xi |$ for the functions $\xi$ belonging to the newly defined class.