In this paper, we introduce and investigate an interesting subclass $\Cal N^{h,p}_\Sigma(\lambda,\mu)$ of analytic and bi-univalent functions in the open unit disk $\Bbb U$. For functions belonging to the class $\Cal N^{h,p}_\Sigma(\lambda,\mu)$, we obtain estimates on the first two Taylor--Maclaurin coefficients $|a_2$ and $|a_3|$. The results presented in this paper would generalize and improve some recent works of \c Ca\u glar et al. [3], Xu et al. [10], and other authors.