  The mixed principal eigenvalue of $p\,$-Laplacian (equivalently, the optimal
constant of weighted Hardy inequality in $L^p$ space) is studied in this paper.
Several variational formulas for the eigenvalue are presented. As applications
of the formulas, a criterion for the positivity of the eigenvalue is obtained.
Furthermore, an approximating procedure and some explicit estimates are
presented case by case. An example is included to illustrate the power of the
results of the paper.
