Conformal $\beta$- change in Finsler spaces

We investigate what we call a conformal $\beta$ - change in Finsler spaces, namely $$L(x,y)\to ~^{\ast}L(x,y)=e^{\sigma(x)}L(x,y)+\beta(x,y)$$ where~$\sigma ~$ is a function of $x~ only ~and ~ \beta (x, y)$ is a given 1- form. This change generalizes various types of changes: conformal changes, Randers changes and $\beta$ - changes. Under this change, we obtain the relationships between some tensors associated with $(M,L)$ and the corresponding tensors associated with $(M,{^\ast}L)$. We investigate some $\sigma$- invariant tensors . This investigation allows us to give an answer to the question: Are the properties of C-reducibility, $S_3$-likeness and $S_4$-likeness invariant under a conformal $\beta$ - change?