The duo Fenchel-Young divergence

By calculating the Kullback-Leibler divergence between two probability measures belonging to different exponential families, we end up with a formula that generalizes the ordinary Fenchel-Young divergence which is recovered in the special case when we let the two exponential families coincide. Inspired by this formula, we define the duo Fenchel-Young divergence and reports a dominance condition on its pair of generators which guarantees that it is always non-negative.