Two optimal inequalities for anti-holomorphic submanifolds and their applications

The CR $delta$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $delta$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR $delta$-invariant. Moreover, we obtain some classification results for certain anti-holomorphic submanifolds in complex space forms which satisfy the equality case of either inequality.