The class of life distributions used better than aged in convex order upper tail ordering (UBACT) is introduced. A Moment inequality to this class (UBACT) of life distribution is given. In addition testing exponentiality versus (UBACT) class of life distribution based on a moment inequality is presented. Simulation such as critical values, Pitmans asymptotic efficiency and the power of test are discussed. Medical applications are given at the end of the paper.