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Comparative convexity is a generalization of convexity relying on abstractnotions of means. We define the Jensen divergence and the Jensen diversity fromthe viewpoint of comparative convexity, and show how to obtain the generalizedBregman divergences as limit cases of skewed Jensen divergences. In particular,we report explicit formula of these generalized Bregman divergences whenconsidering quasi-arithmetic means. Finally, we introduce a generalization ofthe Bhattacharyya statistical distances based on comparative means usingrelative convexity.

Generalizing Jensen and Bregman divergences with comparative convexity and the statistical Bhattacharyya distances with comparable means