Motivated by the works of H. M. Srivastava et al. [7], we introduce and investigate two new general subclasses ${\cal H_T}(\varphi,\psi,\alpha),{\cal H}^{h,p}_{\cal T}(\alpha)$ of bi-starlike and bi-convex of Ma-Minda type functions. Bounds on the first two coefficients $|a_2|$ and $|a_3|$ for functions in ${\cal H_T}(\varphi,\psi,\alpha)$ and ${\cal H}^{h,p}_{\cal T}(\alpha)$ are given. The results here generalize and improve the corresponding earlier works done by Ali et al. [1] and Brannan et al. [2].