  We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K\"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact K\"ahler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are K\"ahler analogues of well-known results for Riemannian manifolds. 
Lower bounds for the first eigenvalue of the Laplacian on K\"ahler   manifolds