Two inequalities for the first Robin eigenvalue of the Finsler Laplacian

Let \Omega be a bounded connected, open set of \R^n with Lipschitz boundary. Let F be a suitable norm in \R^n and let \Delta_F u be the so-colled Finsler Laplacian. In this paper we prove two inequalities for the first eigenvalue of \Delta_F with Robin boundary conditions involving a positive function \beta. As a consequence of our result we obtain the asymptotic behavior of this eigenvalue when \beta is a positive constant which goes to zero.