Abstract: In this talk, we set $(M,g)$ be an open Riemannian manifold, and $\gamma:[0,\infty)\rightarrow M$ a unit-speed ray on $M$. We firstly rieview some estiamtions about $\Ric(\dot\gamma,\dot\gamma)$ along $\gamma$, then we assume the Ricci curvature of $M$ is non-negative and establish some new inequalities about $\Ric(\dot\gamma,\dot\gamma)$ by analysing the Riccati inequality about distance functions on Riemannian manifolds. Based on this,  we give some extensions of the Bonnet-Myers theorem in integral type.