Segre embedding and biharmonicity

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 $M_1^p\times M_2^q$ 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.