Faber polynomials estimates for bi-univalent functions of complex order involving q-derivative

Abstract

In this paper, we define a new subclass of bi-univalent functions of complex order ∑q(τ,ζ;φ) which is defined by subordination in the open unit disc D by using ∇qF(ϑ) operator. Furthermore, using the Faber polynomial expansions, we get upper bounds for the coefficients of function belonging to this class. It is known that the calculus without the notion of limits is called q-calculus which has influenced many scientific fields due to its important applications. The generalization of derivative in q-calculus that is q-derivative was defined and studied by Jackson. A function 𝐹 ∈ 𝐴 is said to be bi-univalent in D if both F and F^-1 are univalent in D. The class consisting of bi-univalent functions is denoted by σ. The Faber polynomials play an important role in various areas of mathematical sciences, especially in geometric function theory. The purpose of our study is to obtain bounds for the general coefficients |an|(n≥3)by using Faber polynomial expansion under certain conditions for analytic bi-univalent functions in subclass ∑q(τ,ζ;ϕ) and also, we obtain improvements on the bounds for the first two coefficients |a2|and|a3| of functions in this subclass. In certain cases, our estimates improve some of those existing coefficient bounds. Keywords: analytic functions; bi-univalent functions; coefficient bounds; subordination; q-derivative; Faber polynomials 2010 Mathematical Subject Classification: 30C45