Laguerre–Samuelson type inequalities

We give a generalization of Laguerre–Samuelson’s inequality in the framework of group theory. This extends a recent result of Trenkler and Puntanen [G. Trenkler, S. Puntanen, A multivariate version of Samuelson’s inequality, Linear Algebra Appl. 410 (2005) 143–149]. Some examples are given.