  A variational expression of the reduced relative entropy is given. A reduced
Tsallis relative entropy is defined and some results are given. In particular,
the convexity of the reduced Tsallis relative entropy is given. An
interpolational inequality between Golden--Thompson and Jensen's trace
inequalitie is given for one--parameter extended exponential function and
positive definite matrices. In addition, a lower bound of the reduced Tsallis
relative entropy is given under a certain assumption, by showing a variational
expression of the reduced Tsallis relative entropy. Finally, an upper bound of
the reduced Tsallis relative entropy is given.
