Retraction Note to: Toeplitz Matrices Whose Elements are the Coefficients of Starlike and Close-to-Convex Functions

Abstract

The authors have retracted this article because the article contains major flaws in the proof of the main results. The results of the paper are invalid, since the assumption that the functionals considered are rotationally invariant is not valid. All authors have agreed to this retraction.

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

“…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.…”

Toeplitz Determinants Whose Elements Are the Coefficients of Analytic and Univalent functions

“…For a summary of applications of Toeplitz matrices to a wide range of areas of pure and applied mathematics, we refer to [12]. Recently, Thomas and Halim [11] introduced the symmetric Toeplitz determinant T q (n) for analytic functions f of the form (1.1), defined by where n, q = 1, 2, 3, . .…”

“…For small values of n and q, estimates of the Toeplitz determinant |T q (n)| for functions in S * and K have been studied in [11]. Similarly, estimates of the Toeplitz determinant |T q (n)| for functions in R have been studied in [8], when n and q are small.…”

“…Similarly, estimates of the Toeplitz determinant |T q (n)| for functions in R have been studied in [8], when n and q are small. Apart from [8] and [11], there appears to be little in the literature concerning estimates of Toeplitz determinants. In [8,11], we observe an invalid assumption in the proofs.…”

“…Apart from [8] and [11], there appears to be little in the literature concerning estimates of Toeplitz determinants. In [8,11], we observe an invalid assumption in the proofs. It is the purpose of this paper to give estimates for Toeplitz determinants T q (n) for functions in S, S * , C, K, R and T , when n and q are small.…”