  In this paper, we successfully generalize the eigenvalue comparison theorem
for the Dirichlet $p$-Laplacian ($1<p<\infty$) obtained by Matei [A.-M. Matei,
First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (8)
(2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the
$p$-Laplacian in a Riemannian manifold, Tokyo J. Math. 21 (1998) 135--140],
respectively. Moreover, we use this generalized eigenvalue comparison theorem
to get estimates for the first eigenvalue of the Dirichlet $p$-Laplacian of
geodesic balls on complete Riemannian manifolds with radial Ricci curvature
bounded from below w.r.t. some point. In the rest of this paper, we derive an
upper and lower bound for the heat kernel of geodesic balls of complete
manifolds with specified curvature constraints, which can supply new ways to
prove the most part of two generalized eigenvalue comparison results given by
Freitas, Mao and Salavessa in [P. Freitas, J. Mao and I. Salavessa, Spherical
symmetrization and the first eigenvalue of geodesic disks on manifolds,
submitted (2012)].
