The study of positive linear operators is closely linked to the study of probability distributions. In this paper, we investigate the relationship between classical positive linear operators, commonly used in Approximation Theory, and the entropies associated with probability distributions. These entropies are special functions, particularly Heun functions, convex functions, and logarithmically convex functions, that satisfy specific equalities and inequalities, for which various bounds can be established. These properties are important for a wide range of applications. Our paper is a (non-exhaustive) survey. We present results without proofs, but with detailed bibliographic references.