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 $\bigl[\frac{m_r}{r!}\bigr]^{r+1}=\bigl[{m_{r+1}}{(r+1)!}\bigr]^r$ for some $r\ge1$, then $F(t)=1-e^{-\lambda t}$; in addition we give upper and lower bounds for the integral $\int_0^\infty e^{-st}[1-F(t)]\,dt$ expressed in terms of $m_1$ and $m_2$.