Using a real-valued functional (denoted by p) introduced by LOZINSEY and defined on matrices, lower and upper bounds can be given for the norm of solutions of linear differential equations. This functional also depends on the norm applied. It is pointed out that the best possible bounds can be obtained applying an appropriate linear transformation of the differential equation. In the real and autonomous case this appropriate linear transformation is real and does not depend on the time if p is

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... nduced by the Euclidean norm. Moreover, close correspondence between p and quadratic Liapunov functions i8 shown. It is proved that in the general case (not Euclidean norms) the best possible bounds can, generally, not be obtained if the linear transformation does not depend on the time.