An inequality for a functional on aging distribution functions

We prove an inequality for a functional on aging distribution functions F(t), which makes it possible to obtain inequalities for \(m_r = \int_0^\infty {t^r } dF (t)\) . We show that if \(\left {\frac{{m_r }}{{r!}}} \right]^{r + 1} = \left {\frac{{m_{r + 1} }}{{(r + 1)!}}} \right]^r \) for some r ≥ 1, then F(t) = 1^?e^?λt; in addition we give upper and lower bounds for the integral \(\int_0^\infty {e^{ - st} } 1 - F(1)] dt,\) expressed in terms of m₁ and m₂.