Abstract: The weighted Myers' theorem gives an upper bound estimate
for the diameter of a complete Riemannian manifold with the
$\tau$-Bakry-\'{E}mery curvature bounded from below by a positive
number. The lower bound estimate for the diameter of a compact
manifold is also an interesting question. In this paper, a gradient
estimate for the potential function of a special
$\tau$-quasi-almost-Einstein metric was established by using the
Hopf's maximum principle. Based on it, a lower bound estimate for
the diameter of this metric was derived. The result generalizes
Wang's lower diameter estimate for compact $\tau$-quasi-Einstein
metrics.

Key words: quasi-almost-Einstein metric, gradient estimate, diameter estimate