An Explicit Bound for the First Dirichlet Eigenvalue via Two Moments

Let $(M^n,g)$, $n\ge2$, be a compact connected Riemannian manifold with smooth nonempty boundary, $\operatorname{Ric}_M\ge(n-1)Kg$ for $K>0$, and nonnegative outward boundary mean curvature. We derive an explicit algebraic lower bound $B_η$ for the first Dirichlet eigenvalue from Kasue's mixed inrad