Toeplitz matrices whose elements are coefficients of Bazilevič functions

Abstract

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.

"Upper bounds of Toeplitz determinants for a subclass of alpha-close-to-convex functions"

“…□ Remark 2.4. Theorem 2.3 for α = 0 gives the bound |T 3 (1)| ≤ 13/9 for the class of functions with bounded boundary rotation R derived by Radhika et.al [15] and in [14] for β = 1.…”

“…Thus the result follows from (2.8), (2.9). Equality in (2.7) holds for the function f α , that [15] and in [14] for β = 1.…”

“…Then ∂F3 ∂µ = 0 implies that c = 2, which is a contradiction. Thus for the maximum of F 3 (c, µ), we need only to consider the end points of [15] and in [14] for β = 1. It would be interesting to know the function f α ∈ Rα so that the equality holds in Theorem 2.4 for α ≤ α 0 .…”

Toeplitz Determinant for a Subclass of Tilted Starlike Functions with Respect to Conjugate Points

“…In recent years, the studies in a problem of estimating the coefficient bounds for the Toeplitz determinants for the class of univalent functions and its subclasses have been done by some researchers such as Al-Khafaji et al (2020), Ali et al (2018, Radhika et al (2018Radhika et al ( , 2016, Ramachandran and Kavitha (2017), , Srivastava et al (2019), Halim (2016), andZhang et al (2019). However, there was no study of finding estimates for the Toeplitz determinants α δ consisting the functions given by (2).…”

Bounds on Toeplitz Determinant for Starlike Functions with Respect to Conjugate Points

“…The concept of Toeplitz matrices led to the development of the studies related to Toeplitz determinants, Toeplitz kernel, Toeplitz operators, and q-deformed Toeplitz matrices [8]. In a recent investigation, the Toeplitz determinant has been studied by [9][10][11][12][13][14][15][16][17][18], and they succeeded in estimating the coefficient bounds for Toeplitz determinant…”