In this paper a
kind of integral operator Iβαf(z) related to the
parameters α,β in an open unit disk is investigated. Firstly,
the bi-univalent function classes Hα,βΣq(λ;ϕ) and Lα,βΣq(μ,λ;ϕ) involving this integral operator and the q-derivative operator are defined by applying the subordination principle of analytic functions. Then, the upper bounds of the first two coefficients a2 and a3 of the two classes of bi-univalent analytic functions are estimated, and the corresponding Fekete-Szegö inequalites for these classes are obtained.