## Variation of the Failure Rate

I have three questions, if anyone can answer anyone of them, I would be indebted to their help.

Q1:
Is there a name for this ratio?
$
r(x) = \dfrac{\int^{\alpha x +d}_{x}f(x)dx}{f(x)},
$

where $\alpha <1$ and $d>0$.

Q2:
Lets focus on $x<\frac{d}{1-\alpha}.$
Does a decreasing $r$ imply an increasing failure (hazard) rate? Or vice versa?
(The failure rate is
$
h(x) = \dfrac{f(x)}{\int^{\infty}_{x}f(x)dx}
$
)

Q3:
Are there any other simpler properties that imply an increasing $r$?

Thank you!