  We study the eigenvalue problem for the $p$-Laplacian on K\"ahler manifolds.
Our first result is a lower bound for the first nonzero eigenvalue of the
$p$-Laplacian on compact K\"ahler manifolds in terms of dimension, diameter,
and lower bounds of holomorphic sectional curvature and orthogonal Ricci
curvature for $p\in (1, 2]$. Our second result is a sharp lower bound for the
first Dirichlet eigenvalue of the $p$-Laplacian on compact K\"ahler manifolds
with smooth boundary for $p\in (1, \infty)$. Our results generalize
corresponding results for the Laplace eigenvalues on K\"ahler manifolds proved
in [14].
