On semi-$C$-reducible $\beta$-conformal change

We consider a $\beta$-conformal change $\bar{F}$ of a Finsler metric $F$, defined as $\bar{F}= e^{\sigma(x)}F + \beta$, where $\sigma(x)$ is a function of $x$ and $\beta(x,y) = b_i(x)y^i$ is a 1-form on $M$. We obtain a relationship between mean Cartan tensor $F$ and $\bar{F}$ when these metrics are semi-$C$-reducible and an example of well-known metrics is provided. This relationship gives different results for metrics with dimension 3 and greater than 3.