Upper and lower bounds for the first Robin eigenvalue of nonlinear elliptic operators

Let $\mathrm{\Omega }$ be a bounded, smooth domain of ${\mathbb{R}}^N$, $N\ge 2$. In this paper, we prove some inequalities involving the first Robin eigenvalue of the $p$-laplacian operator. In particular, we prove an upper bound for the first Robin eigenvalue of nonlinear elliptic operators in terms of the first Dirichlet eigenvalue.