----------------------------------------------------------------------------

As we mentioned in the introduction, the one loop corrections involving the matter fields in the curved space-time generally provide the correction terms with the quadratic and logarithmic in the Ricci scalar [@Nariai73; @DeWitt67]. This action is given in Eq.(\[SR2lnR\]), one can rewrite this as $$\begin{aligned}
 S &= \frac{\Mp^2}{2} \int \sqrt{-g} \, d^4 x R \left(1 + \alpha \frac{R}{M^2} + \beta \frac{R}{M^2} \ln \frac{R}{M^2} \right) \equiv \frac{\Mp^2}{2} \int \sqrt{-g} \, d^4 x M^2 \bar{f}(y) \nonumber \\ 
&= \frac{\bMp^2 M^2}{2} \int \sqrt{-g} \, d^4 x \, y \left(1 + \alpha y + \beta y \ln y \right) \, , \label{Sx2lnx} \end{aligned}$$ where $M$ is the characteristic mass scale can be tuned in order to obtain the proper inflationary behavior, $y = R/M^2$ is the normalized dimensionless variable (curvature), and $\bar{f}$ is a dimensionless function. For the given $f(R)$, one can obtain the analytic form of the Ricci scalar, and the scalar potential, $V(\phi)$ as nicely shown in the reference [@14047349] $$\begin{aligned}
 R &= \frac{e^{\sqrt{\frac{2}{3}} \bphi} -1 }{2 \beta W_{k}(X)} = \frac{\bff_{,y} - 1}{2 \beta W_{k}(X)}  \label{RforR2} \,\,  , \\
X &\equiv \frac{e^{(2\alpha+\beta)/2\beta}}{2\beta} \left( e^{\sqrt{\frac{2}{3}} \bphi} -1 \right) = \frac{e^{(2\alpha+\beta)/2\beta}}{2\beta} \left( \bff_{,y} - 1 \right)   \label{XforR2} \,\, , \\
V(\bphi) &= 