Variance-Refined In-Diameter Lower Bound for the First Dirichlet Eigenvalue

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with nonempty boundary and $n\geq 2$. Assume that ${\mathrm{Ric}(M)\ge (n-1)K}$ for some ${K>0}$ and that $\partial M$ has nonnegative mean curvature with respect to the outward unit normal. Denote by $λ$ the first Dirichlet eigenvalue of