An axiomatic characterization of a two-parameter extended relative entropy

Shannon entropy [1] is one of fundamental quantities in classical information theory and uniquely determinded by the Shannon-Khinchin axiom or the Faddeev axiom. Oneparameter extensions for Shannon entropy have been studied by many researchers [2]. The Rényi entropy [3] and the Tsallis entropy [4] are famous. In the paper [5], the uniqueness theorem for the Tsallis entropy was proven. See also the paper [6] and the references therein, for the axiomatic characterizations of one-parameter extened entropies. Recently, a twoparameter extended entropy was studied by several researchers [7], [8], [9], [10], [11], [12] and the uniqueness theorem for a two-parameter extended entropy was proven in [12] by generalizing the Shannon-Khinchin axiom. A two-parameter extended entropy is defined by