First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere

S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ 1 of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S n+1. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S n+1 with λ 1 = n is either isometric to the sphere S n or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate.