Peierls-Bogolyubov's inequality for deformed exponentials

We study convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and obtain in this way new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls-Bogolyubov's inequality. We use these results to improve previously known lower bounds for the Tsallis relative entropy.