Log-concavity for Bernstein-type operators using stochastic orders

This paper aims to study the preservation of log-concavity for Bernstein-type operators. In particular, attention is focused on positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of a process with independent increments. This class includes the classical Gamma, Szász, and Szász–Durrmeyer operators. With respect to the first and second operators, the results of this paper correct two erroneous counterexamples in [10]. As a main tool in our results we use stochastic order techniques. Our results include, as a particular case, the log-concavity of certain functions related to the incomplete Gamma function.