In the present paper, we introduce and investigate two new subclasses, namely; the class of strongly α–bi-spirallike functions of order β and α–bi-spirallike functions of order ρ, of the function class ∑; of normalized analytic and bi-univalent functions in the open unit disk U={z:z∈ℂ   and   ∣z∣<1}.

We find estimates on the coefficients |a₂|, |a₃| and |a₄| for functions in these two subclasses.

Keywords: univalent functions, bi-univalent functions, bi-spirallike functions, Taylor-Maclaurin series, coeffcient bounds.