Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain

Abstract

In this present paper, we define a new operator in conjugation with the basic (or q-) calculus. We then make use of this newly defined operator and define a new class of analytic and bi-univalent functions associated with the q-derivative operator. Furthermore, we find the initial Taylor–Maclaurin coefficients for these newly defined function classes of analytic and bi-univalent functions. We also show that these bounds are sharp. The sharp second Hankel determinant is also given for this newly defined function class.

Coefficient Inequalities of q-Bi-Univalent Mappings Associated with q-Hyperbolic Tangent Function

“…Moreover, after deducting the Equation labeled as (33) from Equation (30), and taking into account the equalities denoted as (34) and (35), we arrive at the subsequent outcome:…”

“…To derive the coefficient a 3 , we can achieve this by inserting the initial Equation (37) into (35).…”

Concerning a Novel Integral Operator and a Specific Category of Starlike Functions

“…Thereafter, many authors have considered it in the definitions of many subclasses of normalized analytic functions. For instance, see [14][15][16][17][18][19][20][21][22][23][24][25][26] and the comprehensive report on operators by Shareef et al [27], for some specifics.…”

Properties for a Certain Subclass of Analytic Functions Associated with the Salagean q-Differential Operator

“…Based on the same idea, many authors have extensively studied the q-calculus operators (q-differential and q-integral operators) in GFT. A recent study on these operators acting on analytic functions can be found in [12][13][14][15][16][17][18][19]. For 0 < q < 1, Jackson [9,10] defined the q-differential operator, D q , of a function, ξ, as the following:…”