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

The famous Toeplitz matrix is a matrix in which each descending diagonals form left to right is constant, this mean $T=\left(\begin{array}{cccc} a_{0} & a_{1} & \cdots & a_{n} \\ a_{-1} & a_{0} & \cdots & \cdots \\ \vdots & \vdots & \ddots & \vdots \\ \cdots & \cdots & a_{-1} & a_{0} \end{array}\right)$. 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 ∈ : |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.