  Within a framework of utmost generality, we show that the entropy
maximization procedure with linear constraints uniquely leads to the
Shannon-Boltzmann-Gibbs entropy. Therefore, the use of this procedure with
linear constraints should not be extended to the generalized entropies
introduced recently. In passing, it is remarked how the forceful use of the
entropy maximization for the Tsallis and R\'enyi entropies implies either the
Shannon limit of these entropies or self-referential contradictions. Finally,
we note that the utilization of the entropy maximization procedure with
different averaging schemes is beyond the scope of this work.
