For an arbitrary nonempty, open set <sup><i>n</i></sup>, <i>n</i> of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian (-\Delta)^m|_{{C^\infty_0}(\Omega)}, <i>m</i> , and its Kreinvon Neumann extension <i>A</i><sub><i>K</i>,,<i>m</i></sub> in <i>L</i><sup>2</sup>(). with <i>N</i>(, <i>A</i><sub><i>K</i>,,<i>m</i>)</sub>, (-\Delta)^m|_{{C^\infty_0}(\Omega)}, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of <i>A</i><sub><i>K</i>,,<i>m</i></sub>, we derive the bound <i>N</i>(, <i>A</i><sub><i>K</i>,,<i>m</i></sub>) (2<i></i>)<sup>-<i>n</i></sup><i></i><sub>n</sub>||{1 + [2<i>m</i>/(2<i>m</i> + <i>n</i>)]}<sup><i>n</i>/(2<i>m</i>)</sup><sup><i>n</i>/(2<i>m</i>)</sup>, > 0, where <i><sub>n</sub></i> <i></i><sup><i>n</i>/2</sup> /((<i>n</i> + 2)/2) denotes the (Euclidean) volume of the unit ball in <sup><i>n</i></sup>.