  Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive
solutions of the heat equation on closed Riemannian manifolds with the Ricci
curvature bounded below by a negative constant. The proof is based on an
integral iteration argument which utilizes Hamilton's gradient estimate, heat
kernel Gaussian bounds and parabolic Harnack inequality.
  In this paper, we show that the sharp Li-Yau estimate can actually be
obtained directly following the classical maximum principle argument, which
simplifies the proof in [26]. In addition, we apply the same idea to the heat
and conjugate heat equations under the Ricci flow and prove some Li-Yau type
estimates with optimal coefficients.
