AN IMPROVED CHEN-RICCI INEQUALITY FOR LEGENDRIAN SUBMANIFOLDS IN SASAKIAN SPACE FORMS

B.Y. Chen [Arch. Math. (Basel), 74(2000), 154-160] proved a geometrical inequality for Lagrangian submanifolds in complex space forms in terms of the Ricci curvature and the squared mean curvature. Recently, this Chen- Ricci inequality was improved in [Int. Electron. J. Geom., 2(2009), 39-45]. On the other hand, the first author [J. Austral. Math. Soc., 72(2002), 247-256] established a Chen-Ricci inequality for submanifolds, in particular C-totally real submanifolds, in Sasakian space forms. In this article, we improve the above inequality for Legendrian submanifolds in Sasakian space forms. We also investigate the equality case of the inequality