Abstract
In the present work, moments inequalities are derived for the harmonic new better than used in expectation upper tail (HNBUET) which demonstrate that if the mean life is finite for any of them then all moments exist. Next, based on these inequalities, new testing procedure for exponentiality against HNBUET is introduced and studied. This test is simpler than most earlier ones and holds high relative efficiency for some commonly used alternatives. The test also enjoys good power.