Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel <i>q</i>-Differential Operator Associated with <i>q</i>-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.

Keywords

• q-limaçon domain
• Bi-univalent functions
• Hankel determinant
• starlike function
• Fekete–Szegö inequality