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

Is there a name for this ratio?
<br />
r(x) = \dfrac{\int^{\alpha x +d}_{x}f(x)dx}{f(x)},<br />
where \alpha <1 and d>0.

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
<br />
h(x) = \dfrac{f(x)}{\int^{\infty}_{x}f(x)dx}<br />

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

Thank you!