Toeplitz Matrices Whose Elements Are the Coefficients of Functions with Bounded Boundary Rotation

which are analytic in the open unit disk U = {z : |z| < 1} and let S denote the subclass of A consisting of univalent functions. Obviously, for functionsf ∈ S, wemust havef ̸ = 0 in U. For f ∈ S we consider the family R of functions of bounded boundary rotation so that Re(f󸀠(z)) > 0 in U. The familyR is properly contained in the class of close-to-convex functions (e.g., see Brannan [1], Pinchuk [2], or Duren [3] pp. 269–271.) Toeplitz matrices are one of the well-studied classes of structured matrices. They arise in all branches of pure and applied mathematics, statistics and probability, image processing, quantum mechanics, queueing networks, signal processing, and time series analysis, to name a few (e.g., see Ye and Lim [4]). Toeplitz matrices have some of the most attractive computational properties and are amenable to a wide range of disparate algorithms and determinant computations. Here we consider the symmetric Toeplitz determinant


Introduction
Let A denote the class of all functions  of the form which are analytic in the open unit disk U = { : || < 1} and let S denote the subclass of A consisting of univalent functions.Obviously, for functions  ∈ S, we must have   ̸ = 0 in U.For  ∈ S we consider the family R of functions of bounded boundary rotation so that (  ()) > 0 in U.The family R is properly contained in the class of close-to-convex functions (e.g., see Brannan [1], Pinchuk [2], or Duren [3] pp. 269-271.) Toeplitz matrices are one of the well-studied classes of structured matrices.They arise in all branches of pure and applied mathematics, statistics and probability, image processing, quantum mechanics, queueing networks, signal processing, and time series analysis, to name a few (e.g., see Ye and Lim [4]).Toeplitz matrices have some of the most attractive computational properties and are amenable to a wide range of disparate algorithms and determinant computations.Here we consider the symmetric Toeplitz determinant and obtain sharp bounds for the coefficient body |  ()|;  = 2, 3;  = 1, 2, 3, where the entries of   () are the coefficients of functions  of form (1) that are in the family R of functions of bounded boundary rotation.As far as we are concerned, the results presented here are new and noble and the only prior compatible result is published by Thomas and Halim [5] for the classes of starlike and close-to-convex functions.
It is worth noticing that the bounds presented here are much finer than those presented in [5].

Main Results
We note that, for the functions  of form (1) that are in the family R of functions of bounded boundary rotation, we can write   () = ℎ(), where ℎ ∈ P, the class of positive real part function satisfying (ℎ()) > 0 for  ∈ U and ℎ is of the form We shall state the following result [6], to prove our main theorems.
We remark that the sharp bound | 2 3 −  2 2 | ≤ 5/9 given by Theorem 2 is much finer than | 2 3 −  2 2 | ≤ 5 that was obtained by Thomas and Halim [5] for the class of functions of form (1) that are close-to-convex in U.
Next, we determine a sharp bound for the coefficient body Theorem 3. Let  ∈ R be given by (1).Then As before, without loss of generality, we assume that  1 = , where 0 ≤  ≤ 2.Then, by using the triangle inequality and the fact that || ≤ 1, we obtain Considering the modulus as positive, we get One can apply an elementary calculus to show that Ψ() attains its maximum value of 13/9 on [0, 2] when  = 0.
We remark that the sharp bound | 3 (1)| ≤ 13/9 given by Theorem 4 is much finer than | 3 (1)| ≤ 8 obtained by Thomas and Halim [5] for the class of functions of form (1) that are close-to-convex in U. Finally, an upper bound for the coefficient body | 3 (2)| is presented in the following.Theorem 5. Let  ∈ R be given by (1)  No bounds for | 3 (2)| were obtained by Thomas and Halim [5] for the class of functions of form (1) that are closeto-convex in U.