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 pseudo-umbilical hypersurface and prove: what conditions are satisfied by the first nonzero eigenvalue $\lambda_1$ 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 $\lambda_1=n$ is either isometric to the sphere $S^n$ or for this hypersurface an inequality is fulfilled in which sectional curvatures of the hypersuface $M$ participate.