  In this article, we consider Bergman kernels with respect to modules at
boundary points, and obtain a log-subharmonicity property of the Bergman
kernels, which deduces a concavity property related to the Bergman kernels. As
applications, we reprove the sharp effectiveness result related to a conjecture
posed by Jonsson-Must\c{a}t\u{a} and the effectiveness result of strong
openness property of the modules at boundary points.
