Applications of q-Symmetric Derivative Operator to the Subclass of Analytic and Bi-Univalent Functions Involving the Faber Polynomial Coefficients

Abstract

In this paper, using the basic concepts of symmetric $q$ -calculus operator theory, we define a symmetric $q$ -difference operator for $m$ -fold symmetric functions. By considering this operator, we define a new subclass ${\mathrm{ℛ}}_b\left(\phi ,m,q\right)$ of $m$ -fold symmetric bi-univalent functions in open unit disk $\mathcal{U}$ . As in applications of Faber polynomial expansions for $f_m\in {\mathrm{ℛ}}_b\left(\phi ,m,q\right)$ , we find general coefficient $\left|a_{mk+1}\right|$ for $n\ge 4$ , Fekete–Szegő problems, and initial coefficients $\left|a_{m+1}\right|$ and $\left|a_{2m+1}\right|$ . Also, we construct $q$ -Bernardi integral operator for $m$ -fold symmetric functions, and with the help of this newly defined operator, we discuss some applications of our main results. For validity of our result, we have chosen to give some known special cases of our main results in the form of corollaries and remarks.