New Applications of Faber Polynomials and <i>q</i>-Fractional Calculus for a New Subclass of <i>m</i>-Fold Symmetric bi-Close-to-Convex Functions

Abstract

Using the concepts of q-fractional calculus operator theory, we first define a (λ,q)-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient bounds for a newly established class using the Faber polynomial expansion method. In addition, the Faber polynomial method is used to examine the Fekete–Szegö problem and the unpredictable behavior of the initial coefficient bounds of the functions that belong to the newly established class of m-fold symmetric bi-close-to-convex functions. Our key results are both novel and consistent with prior research, so we highlight a few of their important corollaries for a comparison.

Keywords

• analytic functions
• quantum (or <i>q</i>-) calculus
• <i>q</i>-fractional derivative
• close-to-convex functions
• <i>m</i>-fold symmetric functions
• Faber polynomial expansion