  We consider the Segre embedding of the product
$\mathbb{C}P^p\times\mathbb{C}P^q$ into $\mathbb{C}P^{p+q+pq}$ and study the
biharmonicity of $M^p\times\mathbb{C}P^q$ and $M^p_1\times M^q_2$ as
submanifolds of $\mathbb{C}P^{p+q+pq}$, where $M$ and $M_1$ are Lagrangian
submanifolds of $\mathbb{C}P^p$ and $M_2$ is a Lagrangian submanifold of
$\mathbb{C}P^q$. We find two new large classes of biharmonic submanifolds in
complex projective space forms.
