Toeplitz Determinant whose Its Entries are the Coefficients for Class of Non-Bazilevi´c Functions

Abstract

The famous Toeplitz matrix is a matrix in which each descending diagonals form left to right is constant, this mean T = ( a 0 a 1 ⋯ a n a − 1 a 0 ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ ⋯ a − 1 a 0 ) . Mathematician, engineers, and physicists are interested into this matrix for their computational properties and appearances in various areas: C *-dynamical systems [1], dynamical systems [6], operator algebra [2], Pseudospectrum and signal processing [10]. The object of this research is to define a new class Non-Bazilevi´c functions N δ in unit disk ↁ = {z ∈ ℂ: |z| < 1} related to exponential function. As well as, we obtained coefficient estimates and an upper bound for the second and third determinant of the Toeplitz matrix such that the entries these matrix are belong to this class.

Toeplitz Determinants for the Class of Functions with Bounded Turning

“…Proof. Let a function f (z) ∈ G (α, δ) given by (1). Then there exists a function p (z) ∈ P of the form (2) such that [10]…”

Results on Toeplitz Determinants for Subclasses of Analytic Functions Associated to q-Derivative Operator

“…The estimates for Toeplitz determinants T r (n) for functions in S * q and R, when n and r are small have been studied in Ali et al (2018) , Al-Khafaji et al (2020) , Al-shbeil et al (2022 , Buyankara and Çağlar (2023) , Soh et al (2021) , Radhika et al (2018) , Ramachandran and Kavitha (2017) , Ayinla and Bello (2021) , Rasheed et al (2023) , Sivasubramanian et al (2016) , Srivastava et al (2019) , Tang et al (2023) , Tang et al (2021) , Wahid et al (2022), Wanas et al (2023). Motivated by these results, this study aims to find the determinants of Toeplitz determinants T r (n) for functions in S * q and R q , when n and r are small.…”

Coefficient Bounds for Symmetric Subclasses of q-Convolution-Related Analytical Functions

“…Jackson [4,5] was the first to use the qcalculus in various applications and to introduce the q-analogue of the standard derivative and integral operators; see [6][7][8][9][10]. About coefficients' interesting results, see [11][12][13][14][15][16]. The qshifted factorial is defined for λ, q ∈ C and n ∈ N 0 = N ∪ {0} as follows…”