  In this article we consider the following weighted nonlinear eigenvalue
problem for the $g-$Laplacian $$ -\mathop{\text{ div}}\left( g(|\nabla
u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad
\text{ in }\Omega\subset \mathbb{R}^n, n\geq 1 $$ with Dirichlet boundary
conditions. Here $w$ is a suitable weight and $g=G'$ and $h=H'$ are
appropriated Young functions satisfying the so called $\Delta'$ condition,
which includes for instance logarithmic perturbation of powers and different
power behaviors near zero and infinity. We prove several properties on its
spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of
$G$, $H$, $w$ and the normalization $\mu$ of the corresponding eigenfunctions.
  We introduce some new strategies to obtain results that generalize several
inequalities from the literature of $p-$Laplacian type eigenvalues.
