  Given a positive function $F$ on $\mathbb S^n$ which satisfies a convexity
condition, for $1\leq r\leq n$, we define for hypersurfaces in
$\mathbb{R}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r; F}$, a
generalization of the usual $r$-th mean curvature function. We also define
$L_{r; F}$ operator, the linearized operator of the $r$-th anisotropic mean
curvature, which is a generalization of the usual $L_r$ operator for
hypersurfaces in the Euclidean space $\mathbb R^{n+1}$. The Reilly type
inequalities for the first eigenvalue of the $L_{r; F}$ operator have been
proved.
