Sharp estimates of the Toeplitz determinants of certain order for the primary subclasses of univalent functions

The primary object of this paper is to investigate sharp estimate to the Toeplitz determinants of third order for the class of bounded turning functions and fourth order for the class of starlike and convex functions in the open unit disk $\mathbb{D},$ which are the fundamental subclasses of univalent functions. The practical tools applied in the derivation of our main results are the coefficient inequalities for the analytic in $\mathbb{D}$ functions from the Carath\'{e}odory class. The problem of finding sharp estimates to the Toeplitz determinants for the function $f,$ when it is a member of certain subclass of univalent functions is technically difficult in the case when $q = 4$ and $s\in\{1, 2\}$, than that in the case when $q=3$ and $s\in\{1, 2\}.$ Here, in our present investigation, we have successfully derived the sharp bounds of third -order namely $T_{3,2}\big(f\big)$ for the class of Bounded turning functions and fourth-order Toeplitz determinants namely $T_{4,1}\big(f\big)$ and $T_{4,2}\big(f\big)$ for the class of starlike and convex functions. With the motivation of these results, researchers may obtain bounds (sharp) for other classes of analytic functions of higher order Toeplitz determinants.