  For the principal eigenvalue with bilateral Dirichlet boundary condition, the
so-called basic estimates were originally obtained by capacitary method. The
Neumann case (i.e., the ergodic case) is even harder, and was deduced from the
Dirichlet one plus a use of duality and the coupling method. In this paper, an
alternative and more direct proof for the basic estimates is presented. The
estimates in the Dirichlet case are then improved by a typical application of a
recent variational formula. As a dual of the Dirichlet case, the refine problem
for bilateral Neumann boundary condition is also treated. The paper starts with
the continuous case (one-dimensional diffusions) and ends at the discrete one
(birth--death processes). Possible generalization of the results studied here
is discussed at the end of the paper.
