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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 FinslerLaplacian. In this paper we prove two inequalities for the first eigenvalue of\Delta_F with Robin boundary conditions involving a positive function \beta. Asa consequence of our result we obtain the asymptotic behavior of thiseigenvalue when \beta is a positive constant which goes to zero.

Two inequalities for the first Robin eigenvalue of the Finsler Laplacian