are large), the curvature squared term is dominant than the Ricci scalar term and this correction term leads to an effective cosmological constant. Therefore, the early universe went through an inflationary de Sitter era without introducing an inflaton field.

These quantum corrections take a simple form in the case of free, massless, conformally invariant fields. The contributions which all fields make to the geometrical part of the vacuum to vacuum amplitude. These contributions originate in the vacuum polarization, which the background geometry induces, and give rise to nonobservable renormalizations as well as physically real radiative corrections [@DeWitt67] $${\cal L}_{\text{vp}} = \beta R^2 \ln \left(\frac{R}{M^2} \right) \, . \label{Lvacuumpol}$$ Thus, when one regards the origin of the Starobinsky model which includes a quadratic Ricci scalar term in addition to the Hilbert-Einstein action, it is also natural to include the terms quadratic and logarithmic in the Ricci scalar. There have been investigation of the $f(R)$-gravity including the logarithmic correction for several aspects. The viable f(R)-gravity models replacing the dark energy have been considered [@0308176; @0309062; @0512109; @07053199; @07081482; @08031081; @09125474; @13057290]. The logarithmic f(R) models are also considered to investigate the relativistic stars [@13077977]. Applying the logarithmic f(R)-gravity models for the early 