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Let $M$ be an $n$-dimensional hypersurface of $S^{n+1}$.  Let $M$ carry the induced spin structure.  Let $H$ denote the mean curvature of the embedding.  Then the classical Dirac operator on $D_M$ has at least $2^{[n/2]}$ eigenvalues satisfying
$\lambda^2 \le \frac{n^2}{4} + \frac{n^2}{4 vol(M)} \int_M H^2$