Strict Neumann Pólya inequality on simply connected planar Lipschitz domains

Main theorem: for every bounded simply connected planar Lipschitz domain $\Omega$, $|\Omega|\mu_j(\Omega)<4\pi j$ for all $j\ge1$, with $0=\mu_0<\mu_1\le\cdots$ counted with multiplicity; equivalently $N_N(E)>|\Omega|E/(4\pi)$ for $E>0$, where $N_N$ counts $\mu_j<E$ including $\mu_0$. It contains the convex planar case, for which Filonov (CPAM 2025) had only a weaker constant, and the disk case of Filonov, Levitin, Polterovich and Sher (2023). Multiply connected domains, non-Lipschitz domains and higher dimensions remain open. The manuscript is not yet on arXiv; a draft is in the registered snapshot. The companion strict Dirichlet result is a separate entry.

Main theorem: for every bounded simply connected planar Lipschitz domain $\Omega$, $|\Omega|\mu_j(\Omega)<4\pi j$ for all $j\ge1$, with $0=\mu_0<\mu_1\le\cdots$ counted with multiplicity; equivalently $N_N(E)>|\Omega|E/(4\pi)$ for $E>0$, where $N_N$ counts $\mu_j<E$ including $\mu_0$. It contains the convex planar case, for which Filonov (CPAM 2025) had only a weaker constant, and the disk case of Filonov, Levitin, Polterovich and Sher (2023). Multiply connected domains, non-Lipschitz domains and higher dimensions remain open. The manuscript is not yet on arXiv; a draft is in the registered snapshot. The companion strict Dirichlet result is a separate entry.