Three subclasses of analytic and bi-univalent functions are introduced through the use q?Gegenbauer polynomials, which a generalization Gegenbauer polynomials. For falling within these subclasses, coefficient bounds a2 a3 as well Fekete–Szegö inequalities derived. Specializing parameters used in our main results leads to number new results.

The object of the present paper is to introduce and investigate new subclasses of the function class Σ of bi-univalent functions defined in the open unit disk U, involving quasi subordination. The coefficients estimate |a2| and |a3| for functions in these new subclasses are also obtained. c ©2017 All rights reserved.

We introduce and investigate two new subclasses ℳ σ (α, λ) and ℳ σ (β, λ) of meromorphic bi-univalent functions defined on Δ = {z : z ∈ ℂ, 1 < |z | <∞}. For functions belonging to these classes, estimates on the initial coefficients are obtained.

We introduce and investigate an interesting subclass NP(λ,δ)(Σ) (n, β; h) of analytic and bi-univalent functions in the open unit disk U. For functions belonging to the class NP(λ,δ)(Σ) (n, β; h), we obtain estimates on the first two Taylor-Maclaurin coefficients |a₂| and |a₃|.

"In this paper, we investigate bounds of the coefficients for subclass analytic and bi-univalent functions. The results presented in paper would generalize improve some recent works other authors."