On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate

Consider a family of distributions with survival distributions that are log concave and stochastically increasing in a parameter over which it will be mixed. It is shown that a necessary and sufficient condition for a mixture over any such family to have an increasing failure rate is that the mixing distribution has an increasing failure rate. Such observations are given on generalising this result as well as weakening the conditions on Prekopa's Theorem.