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In this article, we consider minimal $L^{2}$ integrals on sublevel sets of aplurisubharmonic function with respect to a module at a boundary point of thesublevel sets, and establish a concavity property of the minimal $L^{2}$integrals. As applications, we obtain a sharp effectiveness result related to aconjecture posed by Jonsson-Musta\c{t}\u{a}, which completes the approach fromthe conjecture to the strong openness property; we also obtain a strongopenness property of the module and a lower semi-continuity property withrespect to the module.

Boundary points, Minimal $L^{2}$ integrals and Concavity property