We give an affirmative answer to one of the questions posed by Bourin regarding a special type of inequality referred to as subadditivity inequalities in the case of the Hilbert-Schmidt and the trace norms. We formulate the solution for arbitrary commuting positive operators, and we conjecture that it is true for all unitarily invariant norms and all commuting positive operators. New related trace inequalities are also presented. 2010 Mathematics subject classification: primary 15A60; secondary 47A30, 47A63, 47B15.