By using integral representations the perturbation expansion and the Bogoli
ubov inequality in nonextensive Tsallis statistics are investigated in a un
ified way.. This procedure extends the analysis performed recently by Lenzi
et al. [Phys. Rev, Lett. 80: 218 (1998)] to the quantum (discrete spectral
case: for q < 1. An example is presented ill order to illustrate the metho
d.