Some Coefficient Problems for Subclasses of Holomorphic Functions in Complex Order Associating Sălăgeăn q-Differential Operator

Abstract

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| < 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.