Toeplitz and Hankel Determinants for Certain Subclasses Associated with Sakaguchi Type Functions

In this work, we establish bounds for Hermitian Toeplitz determinants of orders two and three for distinct subclasses of symmetric starlike functions associated with balloon-, limaçon-, and bean-shaped domains. Using subordination theory, we derive explicit upper and lower bounds for Toeplitz determinants of order two and three for each class. Our results reveal a clear geometric hierarchy: the bean-shaped domain imposes the tightest restrictions, while the limaçon- domain permits the widest variation. The analysis is further extended to 2-fold and 3-fold symmetric functions, for which bounds for the third Hankel determinant are obtained. This work demonstrates how symmetry and domain geometry jointly govern coefficient estimates, offering new insights into the interplay between shape and analytic structure.