This note gives some new Razumikhin-type theorems on pth moment stability of stochastic functional differential equations with Markovian switching (SFDEwMS) by using auxiliary ordinary differential equation. The main results of this note allow the diffusion operator associated with the underlying SFDEwMS of the Lyapunov function along a solution of the system to be not always negative. An example is provided to illustrate the effectiveness of the proposed results.