  Convexity is a key concept in information theory, namely via the many
implications of Jensen's inequality, such as the non-negativity of the
Kullback-Leibler divergence (KLD). Jensen's inequality also underlies the
concept of Jensen-Shannon divergence (JSD), which is a symmetrized and smoothed
version of the KLD. This paper introduces new JSD-type divergences, by
extending its two building blocks: convexity and Shannon's entropy. In
particular, a new concept of q-convexity is introduced and shown to satisfy a
Jensen's q-inequality. Based on this Jensen's q-inequality, the Jensen-Tsallis
q-difference is built, which is a nonextensive generalization of the JSD, based
on Tsallis entropies. Finally, the Jensen-Tsallis q-difference is charaterized
in terms of convexity and extrema.
