Log concavity preservation by beta operator based on probability tools

Abstract

A number of papers have dealt with the preservation of log convexity and log concavity based on various operators. For instance, in Badía and Sangüesa [10], the preservation of log convexity and log concavity under Bernstein operators was discussed based on some characteristics of a stochastic process. However, regarding beta-type operators, the preservation of log concavity does not hold based on such a probabilistic method used in Badía and Sangüesa [10]. In this study, we explore the preservation of log concavity for the beta operator using alternative probabilistic tools. Notably, we show results on the preservation of log concavity for monotone log concave functions. Further, some results of application to some specific functions, ageing classes of the deterioration Dirichlet process, related operators and order statistics are provided.