We consider a probability distribution p0(x),p1(x),… depending on a real parameter x. The associated information potential is S(x):=∑kpk2(x). The Rényi entropy and the Tsallis entropy of order 2 can be expressed as R(x)=−logS(x) and T(x)=1−S(x). We establish recurrence relations, inequalities and bounds for S(x), which lead immediately to similar relations, inequalities and bounds for the two entropies. We show that some sequences Rn(x)n≥0 and Tn(x)n≥0, associated with sequences of classical positive linear operators, are concave and increasing. Two conjectures are formulated involving the information potentials associated with the Durrmeyer density of probability, respectively the Bleimann–Butzer–Hahn probability distribution.

We investigate the Choquet boundary for subspaces of parabolic functions and for linearly separating subspaces of continuous functions. The relation of the Choquet boundary with the set of peak points is also investigated.