  The aim of this paper is to obtain optimal estimates for the first Robin
eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*}
\lambda_1(\beta,\Omega)=\min_{\psi\in W^{1,p}(\Omega)\setminus\{0\} }
\frac{\displaystyle\int_\Omega F(\nabla \psi)^p dx +\beta
\displaystyle\int_{\partial\Omega}|\psi|^pF(\nu_{\Omega}) d\mathcal H^{N-1}
}{\displaystyle\int_\Omega|\psi|^p dx}, \end{equation*} where
$p\in]1,+\infty[$, $\Omega$ is a bounded, mean convex domain in $\mathbb
R^{N}$, $\nu_{\Omega}$ is its Euclidean outward normal, $\beta$ is a real
number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates
we found are in terms of the first eigenvalue of a one-dimensional nonlinear
problem, which depends on $\beta$ and on geometrical quantities associated to
$\Omega$. More precisely, we prove a lower bound of $\lambda_{1}$ in the case
$\beta>0$, and a upper bound in the case $\beta<0$. As a consequence, we prove,
for $\beta>0$, a lower bound for $\lambda_{1}(\beta,\Omega)$ in terms of the
anisotropic inradius of $\Omega$ and, for $\beta<0$, an upper bound of
$\lambda_{1}(\beta,\Omega)$ in terms of $\beta$.
