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

Let [Formula: see text] be a compact [Formula: see text]-dimensional Riemannian manifold with nonempty boundary. Assume that [Formula: see text] for some [Formula: see text] and that [Formula: see text] has nonnegative mean curvature with respect to the outward unit normal. Denote by [Formula: see text] the first Dirichlet eigenvalue of the Laplacian. Ling’s gradient-comparison method [4] provides an explicit lower bound for [Formula: see text] in terms of [Formula: see text] and the in-diameter [Formula: see text] (twice the maximal distance from a point of [Formula: see text] to [Formula: see text]). We isolate the only step in Ling’s argument that loses quantitative information: a Jensen–Hölder averaging that replaces a nonconstant one-dimensional comparison function by its mean. Using the uniform strong convexity of [Formula: see text] on [Formula: see text], we refine this averaging by a variance term and thereby retain part of the discarded oscillation. This yields an explicit closed-form in-diameter bound that is strictly stronger than Ling’s estimate for every [Formula: see text], without taking a maximum with the curvature-only Lichnerowicz–Reilly bound.