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Let M be a compact hypersurface with constant mean curvature in Sn+1.Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constant γ(n)depending only on n such that if |H|≤γ(n)and β(n,H)≤S≤β(n,H)+n/18,then S ≡ β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n3/2(n-1)H2+2(n-2)/(n-1)√n2H4+4(n-1)H2.