The Sharp Bounds of Hankel Determinants for the Families of Three-Leaf-Type Analytic Functions

Abstract

The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern mathematical physics, the theory of partial differential equations, engineering, and electronics. In our present investigation, two subfamilies of starlike and bounded turning functions associated with a three-leaf-shaped domain were considered. These classes are denoted by BT3l and S3l*, respectively. For the class BT3l, we study various coefficient type problems such as the first four initial coefficients, the Fekete–Szegö and Zalcman type inequalities and the third-order Hankel determinant. Furthermore, the existing third-order Hankel determinant bounds for the second class will be improved here. All the results that we are going to prove are sharp.

On Sharp Estimate of Third Hankel Determinant for a Subclass of Starlike Functions

“…, which was proved by Gandhi et al in [26] along with (28), ( 29) and (31), we obtained the required result.…”

“…For n = 4, we obtain the function class S * 3L connected with a three-leaf shaped domain which has been studied in [28,29]. For n = 5, it reduces to…”

Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain

“…By employing similar techniques, Khalil Ullah et al [46] and Lecko et al [47] derived the sharp bounds for |H 3,1 (g)| when considering functions belonging to the families S * tanh and S * (1/2), respectively. Additionally, the works of authors [48][49][50][51][52][53] proved sharp bounds for the third-order Hankel determinant in various novel subfamilies of univalent functions. In the present work, we consider a family BT car of bounded turning functions related to the cardioid-shaped domain.…”

The Sharp Bound of the Third Hankel Determinant for the Inverse of Bounded Turning Functions

“…in the unit disc D, were proved by Kowalczyk et al [14][15][16][17] and derived the bounds as 4, 1/4, 4/9 and 4/135 respectively. Some more results on sharp bound on third Hankel determinant for different subclass of an analytic functions are obtain by many authors (see [18][19][20][21][22][23][24][25][26]). Very recently, Rath et al [27] estimated the sharp bound of the third Hankel determinants for the inverse of starlike functions with respect to symmetric points.…”

“…Now, considering the eight edges of the parallelepiped. (i) For x = 1 and y = 0; x = 1 and y = 1 in(19) we have c = 0 and y = 0.…”