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In this paper we derive conditions upon the nonnegative random variable under which the inequality $Dg\left(\xi \right)\le cE{\left[g^{\text{'}}\left(\xi \right)\right]}^2\xi$ holds for a fixed nonnegative constant $c$ and for any absolutely continuous function $g$. Taking into account the characterization of a Gamma distribution we consider the functional $U_{\xi }={sup}_g\frac{Dg\left(\xi \right)}{E{\left[g^{\text{'}}\left(\xi \right)\right]}^2\xi }$ and establishing some of its properties we show that $U_{\xi }\ge 1$ and that $U_{\xi }=1$ iff the random variable $\xi$ has a Gamma distribution.

A characteriyation of the Gamma distribution in terms of the $k$-th conditional moment presented in this paper extends the result of Shunji Osaki and Xin-xiang Li (1988).

Let $\lambda$ denote the failure rate function of the $d,f$. $F$ and let ${\lambda }_1$ denote the failure rate function of the mean residual life distribution. In this paper we characterize the distribution functions $F$ for which ${\lambda }_1=c\lambda$ and we estimate $F$ when it is only known that ${\lambda }_1/\lambda$ or ${\lambda }_1-c\lambda$ is bounded.