A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound

Langford and Laugesen conjectured a sharp two-disk bound for the third Neumann eigenvalue under an upper Gaussian-curvature bound (Math. Ann. 386 (2023), 2255–2281, Conjecture 1.4). We prove the conjectured bound for bounded Lipschitz membranes with the weight regularity used by Langford and Laugesen, and strengthen it to a sharp reciprocal inequality. Let $\mathrm{\Omega }\subset \mathbb{C}$ be a bounded simply connected Lipschitz domain, let $\omega \in C^2(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})$ be positive on $\bar{\mathrm{\Omega }}$, and equip $\mathrm{\Omega }$ with $g=\omega \mathrm{\mid }dz{\mathrm{\mid }}^2$. Suppose $K_g\le K$, $A={\int }_{\mathrm{\Omega }}\omega dx>0$, and $KA<4\pi$ when $K>0$. Enumerate the Neumann eigenvalues, counting multiplicity, by $0={\lambda }_0<{\lambda }_1\le {\lambda }_2\le \cdots$. If $D_K(A\mathrm{/}2)$ is the constant-curvature geodesic disk of area $A\mathrm{/}2$, then $\frac{1}{{\lambda }_2(\mathrm{\Omega },g)}+\frac{1}{{\lambda }_3(\mathrm{\Omega },g)}>\frac{2}{{\lambda }_1(D_K(A\mathrm{/}2))}$ and ${\lambda }_2(\mathrm{\Omega },g)<{\lambda }_1(D_K(A\mathrm{/}2))$. No simplicity of ${\lambda }_1$ or boundary differentiability of $\omega$ is required. The same conclusions hold for relatively compact disk-type Lipschitz domains in smooth Riemannian surfaces. Both bounds are sharp at fixed area and curvature upper bound: for each admissible $K,A$, a sequence of smooth connected domains of area $A$ in the constant-curvature model has fixed-index Neumann spectra converging to those of $D_K(A\mathrm{/}2)\bigsqcup D_K(A\mathrm{/}2)$. Neither extremal value is attained in either connected class. The proof uses two-pole Green coordinates, a positive-kernel comparison, simultaneous centering of two complex moments, and a shifted reciprocal variational estimate.