Variational representations related to Tsallis relative entropy

We develop variational representations for the deformed logarithmic and exponential functions and use them to obtain variational representations related to the quantum Tsallis relative entropy. We extend Golden-Thompson’s trace inequality to deformed exponentials with deformation parameter $q\in [0,1],$ thus complementing the second author’s previous study of the cases with deformation parameter $q\in [1,2]$ or $q\in [2,3]\mathrm{.}$