  We prove that the bilinear maximal Bochner-Riesz operator $T_*^\lambda$ is
bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $L^p(\mathbb
R^n)$ for appropriate $(p_1,p_2,p)$ when $\lambda>(4n+3)/5$.
