On the constant-roll inflation with large and small $eta_H$

We study the seemingly duality between large and small $eta_H$ for the constant-roll inflation with the second slow-roll parameter $eta_H$ being a constant. In the previous studies, only the constant-roll inflationary models with small $eta_H$ are found to be consistent with the observations. The seemingly duality suggests that the constant-roll inflationary models with large $eta_H$ may be also consistent with the observations. We find that the duality between the constant-roll inflation with large and small $eta_H$ does not exist because both the background and scalar perturbation evolutions are very different. By fitting the constant-roll inflationary models to the observations, we get $-0.016leeta_Hle-0.0078$ at the 95% C.L if we take $N=60$ for the models with increasing $epsilon_H$ in which inflation ends when $epsilon_H=1$, and $3.0135le eta_Hle 3.021$ at the 68% C.L., and $3.0115le eta_Hle 3.024$ at the 95% C.L. for the models with decreasing $epsilon_H$.