  The paper suggests the concepts of an upper entropy and a lower entropy. We
propose a new axiomatic definition, namely, upper entropy axioms, inspired by
axioms of metric spaces, and also formulate lower entropy axioms. We also
develop weak upper entropy axioms and weak lower entropy axioms. Their
conditions are weaker than those of Shannon-Khinchin axioms and Tsallis axioms,
while these conditions are stronger than those of the axiomatics based on the
first three Shannon-Khinchin axioms for superstatiscs. The expansibility,
subadditivity and strong subadditivity of entropy are obtained in the new
axiomatics. Tsallis statistics is a special case of the superstatistics which
satisfies our axioms. Moreover, different forms of information measures, such
as Shannon entropy, Daroczy entropy, Tsallis entropy and other entropies, can
be unified under the same axiomatic framework.
