Generalization of Kim's estimates in terms of the trace of divergence-free symmetric tensor

In this paper, we generalized E.C. Kim's estimates by taking into account the trace of the divergence-free symmetric nonzero tensor. We have also shown that E.C. Kim's estimates still valid in the case of the trace of the divergence-free symmetric tensor vanished identically. In the equality case, we characterized eta-Killing spinor with Killing pair over Sasakian spin manifolds.