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In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold  M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere  S n . We also obtain Simons-type inequality for same ambient space form.

Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms