  In this paper, we study the first eigenvalue of Jacobi operator on an
$n$-dimensional non-totally umbilical compact hypersurface with constant mean
curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound
for the first eigenvalue of Jacobi operator, which only depends on the mean
curvature $H$ and the dimension $n$.
