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 lambda,q-fractional differintegral operator Dq lambda:A -> A. Using the lambda,q-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass S*q,beta,lambda of starlike functions of order beta and the class C Sigma lambda,q alpha of bi-close-to-convex functions of order beta. We explore the results on coefficient inequality and Fekete-Szeg & ouml; problems for functions belonging to the class S*q,beta,lambda. 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 beta. We also investigate the erratic behavior of the initial coefficients in the class C Sigma lambda,q alpha of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research.