It was proved in Chen and Dillen (J Math Anal Appl 379(1), 229–239, 2011) and Chen et al. (Differ Geom Appl 31(6), 808–819, 2013) that every Lagrangian submanifold M of a complex space form M~ n(4 c) with constant holomorphic sectional curvature 4c satisfies the following optimal inequality: δ(2,…,2)≤n2(2n-k-2)2(2n-k+4)H2+n2-n-2k2c,where H2 is the squared mean curvature and δ(2 , ⋯ , 2) is a δ-invariant on M introduced by the first author, and k is the multiplicity of 2 in δ(2 , ⋯ , 2) , where n≥ 2 k+ 1. This optimal inequality improves an earlier inequality obtained by the first author in Chen (Jpn J Math 26(1), 105–127, 2000). The main purpose of this paper is to study Lagrangian submanifolds of M~ n(4 c) satisfying the equality case of the optimal inequality (A)