  Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of
the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$,
the dimension of $M$, the Busemann-Hausdorff volume and the reversibility
constant of $(M,F)$.
  For a Randers manifold $(M,F:=\sqrt{g}+\beta)$, where $g$ is a Riemannian
metric on $M$ and $\beta$ an appropriate $1$-form on $M$, it is shown that the
first eigenvalue $\lambda_{1,p}(M,F)$ of the Finslerian $p$-Laplacian defined
by the Finsler metric $F$ is controled by the first eigenvalue
$\lambda_{1,p}(M,g)$ of the Riemannian $p$-Laplacian defined on $(M,g)$.
  Finally, the Cheeger's inequality for Finsler Laplacian is extended for
$p$-Laplacian, with $p > 1$.
