Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

Let (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant alpha>0, and the Ricci curvature satisfies Ric <=(n-1)alpha delta, delta >= 1, then it is shown that either M is isometric to the n-sphere S-n (alpha) or else each nonzero eigenvalue lambda of the Laplacian acting on the smooth Junctions of M satisfies the following: lambda(2)+3n alpha(delta-2)lambda+2n alpha(2)delta(1+(n-1)delta)>0.