of $\alpha$ at each $R$. Thus, for the negative value of $\beta$, the model has the upper bound on the $\phi$ ([*i.e. UV incomplete*]{}). When $\beta = 0$, the model is identical to the Starobinsky model and show the plateau region to have the enough number of e-foldings. When the $\beta$ is positive, the potential becomes unstable with a runaway direction for the large values of $\phi$. In this case, the hilltop type model is realized when $\phi$ rolls towards the origin and $\beta$ value will be constrained in order to get the enough e-folding number.

![image](Fig1.pdf){width="1\linewidth"}

The detail evolution of the inflation model given in the action Eq.(\[Sx2lnx\]) is investigated for the different values of coupling constant $\beta$. These results are obtained from Eqs.(\[bphi\]), (\[V\]), and (\[bphiEq\]). We use $M = 6 \times 10^{12}$ GeV and $\alpha = 1$. In order to obtain the enough number of e-foldings, one needs to tune the initial values of the scalar field, $\bphi_{i}$. For an illustration, we show a typical evolutionary behavior of $\phi$ and $V(\phi)$ in Fig.\[fig-2\]. In this figure, $\beta = 0$ and N-efolings are fixed at 60. In this model, $\bphi_{i}$ is chosen to be 41.36 in order to 