Symmetric Toeplitz Matrices for a Class of New Subclass Functions and Bazilevič Functions

In this work, we describe and investigate a novel collection of analytic functions, including the new functions and the Bazilevič functions. An important component of analytic functions, Bazilevič functions have numerous uses in both pure and practical mathematics. For this class of functions, we concentrate on building Toeplitz matrices, examining their structural characteristics, and evaluating their eigenvalues and trends. We utilise these studies to draw sophisticated mathematical conclusions on the stability and convergence characteristics of Bazilevič functions, as well as possible uses in geometry and differential equations. This work aims to determine coefficient estimates for the functions in this family for the first four determinants symmetric𝒯2(2),𝒯2(3),𝒯3(2), and 𝒯3(1) of the Toeplitz matric. We have also give an example to illustrate the spectrum diagram for some different cases and discuss the monotonicity of the generating function for Toplitz matrices. We also look into a few specific circumstances and the implications of our findings.