In this paper, we introduce and investigate an interesting subclass $\mathcal{B}_{\Sigma }^{h,p}$ of analytic and bi-univalent functions in the open unit disk $\mathbb{U}$. For functions belonging to the class $\mathcal{B}_{\Sigma}^{h,p}$ 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 work of Brannan and Taha \cite{BT}.