Coefficient estimates for a certain subclass of analytic and bi-univalent functions

For λ ≥ 0 and 0 ≤ α < 1 < β, we denote by K (λ;α, β) the class of normalized analytic functions satisfying the two sided-inequality α << ( z f ′(z) f (z) + λ z2 f ′′(z) f (z) ) < β (z ∈ U), where U is the open unit disk. Let KΣ(λ;α, β) be the class of bi-univalent functions such that f and its inverse f −1 both belong to the class K (λ;α, β). In this paper, we establish bounds for the coefficients, and solve the Fekete-Szegő problem, for the classK (λ;α, β). Furthermore, we obtain upper bounds for the first two Taylor-Maclaurin coefficients of the functions in the classKΣ(λ;α, β).