Conditions on unimodality and logconcavity for densities of coherent systems with an application to Bernstein operators

In this note, the distribution of the lifetime of a coherent system with independent and identically distributed component lifetimes is considered. Conditions yielding unimodality or logconcavity of the density function of the system lifetime are obtained. In the conditions, only assumptions on the density function of the components and on the signature of the system are imposed. The results are illustrated with several examples. Additionally, a problem on preservation of logconcavity under the classical Bernstein operator is solved.