We consider the Toeplitz matrices whose elements are the coefficients of Bazilevič functions and obtain upper bounds for the first four determinants of these Toeplitz matrices. The results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation.

In the current exploration, we defined new subclasses of analytic functions, namely $R_{lim}(l,\nu )$ and $C_{lim}(l,\nu )$, defined by subordination linked with a Limacon-shaped domain. We found a few initial coefficient bounds and Fekete–Szegő inequalities for the functions in the above-stated new classes. The corresponding results have been derived for the function $h^{-1}$. Additionally, we discuss the Poisson distribution as an application of our consequences.

The objective of the present article is to introduce new subclasses of bi-Bazilevič functions, bi-quasi-convex functions and $\alpha$-exponentially bi-convex functions involving functions with bounded boundary rotation of order $\nu$. For the above-said newly defined classes, we obtain first two initial coefficient bounds. In addition, the familiar Fekete–Szegö coefficient inequality is too found for the newly introduced subclasses of bi-univalent functions. Apart from the new findings that are obtained, it also improves the prior estimates that are presented already in the literature.