  In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to
the Maxwell-Bolzmann distribution $p_i$ by means of a twofold link. The first
link is differential and is offered by the Jaynes Maximum Entropy Principle.
The second link is algebraic and imposes that both the entropy and the
distribution must be expressed in terms of the same function in direct and
inverse form. Indeed, the Maxwell-Boltzmann distribution $p_i$ is expressed in
terms of the exponential function, while the Boltzmann-Shannon entropy is
defined as the mean value of $-\ln(p_i)$. In generalized statistical mechanics
the second link is customarily relaxed. Here we consider the question if and
how is it possible to select generalized statistical theories in which the
above mentioned twofold link between entropy and the distribution function
continues to hold, such as in the case of ordinary statistical mechanics.
Within this scenario, there emerge new couples of direct-inverse functions,
i.e. generalized logarithms $\Lambda(x)$ and generalized exponentials
$\Lambda^{-1}(x)$, defining coherent and self-consistent generalized
statistical theories. Interestingly, all these theories preserve the main
features of ordinary statistical mechanics, and predict distribution functions
presenting power-law tails. Furthermore, the obtained generalized entropies are
both thermodynamically and Lesche stable.
