  The aim of this paper is to investigate the q -> 1/q duality in an
information-entropy theory of all q-generalized entropy functionals (Tsallis,
Renyi and Sharma-Mittal measures) in the light of a representation based on
generalized exponential and logarithm functions subjected to Kolmogorov's and
Nagumo's averaging. We show that it is precisely in this representation that
the form invariance of all entropy functionals is maintained under the action
of this duality. The generalized partition function also results to be a scalar
invariant under the q -> 1/q transformation which can be interpreted as a
non-extensive two dimensional Ising model duality between systems governed by
two different power law long-range interactions and temperatures. This does not
hold only for Tsallis statistics, but is a characteristic feature of all
stationary distributions described by q-exponential Boltzmann factors.
