Let $r_F $ be the hazard rate function associated with the distribution function F. Let F be a distribution with support in $(0,1]$. F is said to be ${\text{IHR}}( G )( {\text{DHR}}( {\text{G}}) )$ if $\smallint _0^1 r_F ( tu )dG( u )$ is nondecreasing (nonincreasing) in $t\geqq 0$. Some distributions G for which ${\text{IHR}} \Rightarrow {\text{IHR}}( G ) \Rightarrow {\text{IHRA}}$ are identified. Pairs of distributions $G_1 $ and $G_2 $ such that ${\text{IHR}}( G_1 ) \Rightarrow {\text{IHR}}( G_2 )$ are shown. Bounds on ${\text{IHR}}(G)$ distributions are given. It is shown that only exponential distributions can be both ${\text{IHR}}(G)$ and ${\text{DHR}}(G)$. Closure of the class of ${\text{IHR}}(G)$ distributions under formation of series structures is demonstrated.