Remark on a diameter bound for complete Riemannian manifolds with positive Bakry–Émery Ricci curvature

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry–Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (2009) [12] and Limoncu (2012) [8]. As an application, we shall give an upper diameter bound for compact shrinking Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient conditions for four-dimensional compact shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.