A novel function for modified gravity is proposed, $f(R,T)=R+\lambda R^2+2\beta ln\left(T\right)$, with constants $\lambda$ and $\beta$, scalar curvature $R$, and the trace of stress energy tensor $T$, satisfying $T=\rho -3p>0$. Subsequently, two equations of state (EoS) parameters, namely $\omega$ and a parametric form of the Hubble parameter $H$, are employed in order to study the accelerated expansion and initial cosmological bounce of the corresponding universe. Hubble telescope experimental data for redshift $z$ within the range $0.07\le z\le 2.34$ are used to compare the theoretical and observational values of the Hubble parameter. Moreover, it is observed that all the energy conditions are fulfilled within a neighborhood of the bouncing point $t=0$, what shows that the necessary condition for violation of the null energy condition, within a neighborhood of the bouncing point in general relativity, could be avoided by modifying the theory in a reasonable way. Furthermore, a large amount of negative pressure is found, which helps to understand the late time accelerated expansion phase of the universe.

提出了一种新的重力修正函数， $F（\mathrm{[R}，Ť）=\mathrm{[R}+\lambda {\mathrm{[R}}^2+2\beta ln（Ť）$，带有常量 $\lambda$ 和 $\beta$，标量曲率 $\mathrm{[R}$，以及应力能张量的踪迹 $Ť$，令人满意 $Ť=\rho -3p>0$。随后，两个状态方程（EoS）参数，即$\omega$ 和哈勃参数的参数形式 $H$用来研究相应宇宙的加速膨胀和初始宇宙学反弹。哈勃望远镜的红移实验数据$ž$ 在范围内 $0。07\le ž\le 2。34$用于比较哈勃参数的理论值和观测值。此外，观察到所有能量条件都在反弹点附近满足$Ť=0$这表明，通过合理地修改理论，可以避免在广义相对论的弹跳点附近避免违反零能量条件的必要条件。此外，发现了大量的负压，这有助于了解宇宙加速膨胀阶段的后期。