  In this note, we obtain a sharp volume estimate for complete gradient Ricci
solitons with scalar curvature bounded below by a positive constant. Using
Chen-Yokota's argument we obtain a local lower bound estimate of the scalar
curvature for the Ricci flow on complete manifolds. Consequently, one has a
sharp estimate of the scalar curvature for expanding Ricci solitons; we also
provide a direct (elliptic) proof of this sharp estimate. Moreover, if the
scalar curvature attains its minimum value at some point, then the manifold is
Einstein.
