Abstract
This approach finds new upper bounds for the first positive eigenvalue of the p-Laplacian operator using the mean and constant sectional curvatures on Riemannian manifolds. In particular, we provide several estimates for the first nonzero eigenvalue of the p-Laplacian operator on closed orientated totally real submanifolds of dimension m in a generalized complex space form M-n(kappa, epsilon). Moreover, we generalize the Reillyinequality of Laplacian (Reilly in Comment Math Helv 52(4):525-533, 1977) to the p-Laplacian for totally real submanifold in complex projective space and complex Euclidean space for kappa = 1 and kappa = 0, as applications, respectively.