\tilde{\beta}$. This action is also naturally obtained when one considers the small correction in the Starobinsky model $ R + \alpha R^2 + R^{2+ \gamma}$ with $\gamma \ll 1$ as shown in [@14116010]. The inflationary behavior of this action was nicely investigated analytically by using the approximation [@14047349]. We improve the previous work numerically without using any approximation. We also investigate the reheating process in this model. Similar forms of the logarithmic model based on the stability and the viability of observation are also considered [@151209324; @160508016]. Constraints on the model parameters given in this model should be obtained from the CMB observation [@150202114].

Higgs Starobinsky inflation models have been investigated to solve the large Higgs values in the early universe[@150908882; @160502236; @160906887; @161104932; @170106636; @170107665; @170505638; @170700984]. In these models, non-minimal coupling of the Higgs field to the Ricci scalar induces the large quantum corrections. These approaches might explain both the dark matter and the inflation.

In the usual inflation models, soon after the end of the inflation, the inflaton fields begin to oscillate around the minimum of their effective potentials, producing particles, which interact with each other to reach the thermal equilibrium at the reheating temperature, $T_{\text{rh}}$ [@10012600]. 