RETRACTED ARTICLE: Toeplitz Matrices Whose Elements are the Coefficients of Starlike and Close-to-Convex Functions

Third-Order Hankel and Toeplitz Determinants for Starlike Functions Connected with the Sine Function

“…Now, substituting Equations (3), (8), (9) and (10) into Equation 14, we easily obtain the desired assertion (Equation (13)).…”

“…On the other hand, Thomas and Halim [10] defined the symmetric Toeplitz determinant T q (n) as follows:…”

“…In recent years, many authors studied the second-order Hankel determinant H 2 (2) and the third-order Hankel determinant H 3 (1) for various classes of functions (the interested readers can see, for instance, [12][13][14][15][16][17][18][19][20][21][22][23][24][25]). However, apart from the work in [10,21,26,27], there appears to be little literature dealing with Toeplitz determinants. Inspired by the aforementioned works, in this paper, we aim to investigate the third-order Hankel determinant H 3 (1) and Toeplitz determinant T 3 (2) for the above function class S * s associated with sine function, and obtain the upper bounds of the above determinants.…”

Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

“…It is a known fact from [22] that |a n+1 | − |a n | < c for a constant c. However, the problem of finding exact values of the constant c for S and its various subclasses has proved to be difficult. In a very recent investigation, Thomas and Abdul-Halim [23] succeeded in obtaining some sharp estimates for T j (n) for the first few values of n and j involving symmetric Toeplitz determinants whose entries are the coefficients a n of starlike and close-toconvex functions.…”

A Study of Fourth-Order Hankel Determinants for Starlike Functions Connected with the Sine Function

“…In recent years, many papers have been devoted to finding upper bounds for the second-order Hankel determinant H 2 ð2Þ and the third-order Hankel determinant H 3 ð1Þ, whose elements are various classes of analytic functions; it is worth mentioning that [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. For instance, Murugusundaramoorthy and Bulboacă [21] defined a new subclass of analytic functions ML a c ðλ, ϕÞ and got upper bounds for the Fekete-Szegö functional and the Hankel determinant of order two for f ∈ ML a c ðλ, ϕÞ: Islam et al [22] examined the q-analog of starlike functions connected with a trigonometric sine function and discussed some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, and convolution results with partial sums for this class.…”