Three measures of divergence between vectors in a convex set of an-dimensional real vector space are defined in terms of certain types of entropy functions, and their convexity property is studied. Among other results, a classification of the entropies of degree\alphais obtained by the convexity of these measures. These results have applications in information theory and biological studies.

In an earlier work, the authors introduced a divergence measure, called the first-order Jensen difference, or in short\cal j-divergence, which is based on entropy functions of degree\alpha. This provided a generalization of the measure of mutual information based on Shannon's entropy (corresponding to\alpha = 1). It was shown that the first-order\cal j-divergence is a convex function only when a is restricted to some range. We define higher order Jensen differences and show that they are convex functions only when the underlying entropy function is of degree two. A statistical application requiring the convexity of higher order Jensen differences is indicated.