Fractional Differential Operator Based on Quantum Calculus and Bi-Close-to-Convex Functions

In this article, we first consider the fractional q -differential operator and the λ , q -fractional differintegral operator D q λ : A → A . Using the λ , q -fractional differintegral operator, we define two new subclasses of analytic functions: the subclass S * q , β , λ of starlike functions of order β and the class C Σ λ , q α of bi-close-to-convex functions of order β . We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class S * q , β , λ . Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order β . We also investigate the erratic behavior of the initial coefficients in the class C Σ λ , q α of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research.