  The aim of this paper is give a simple proof of some results in \cite{Jun
Ling-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in the
sharp lower bound of the first eigenvalue in the Laplacian operator on compact
Riemannian manifolds with nonnegative Ricci curvature. We also get a result
about lower bound of the first Neumann eigenvalue in a special case. Indeed,
our estimate of lower bound in the this case is optimal. Although the methods
used in here due to \cite{Jun Ling-2006-IJM} (or \cite{JunLing-2007-AGAG}) on
the whole, to some extent we can tackle the singularity of test functions and
also simplify greatly much calculation in these references. Maybe this provides
another way to estimate eigenvalues.
