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?

$\displaystyle

r(x) = \dfrac{\int^{\alpha x +d}_{x}f(x)dx}{f(x)},

$

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

Q2:

Lets focus on $\displaystyle x<\frac{d}{1-\alpha}.$

Does a decreasing $\displaystyle r$ imply an increasing failure (hazard) rate? Or vice versa?

(The failure rate is

$\displaystyle

h(x) = \dfrac{f(x)}{\int^{\infty}_{x}f(x)dx}

$)

Q3:

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

Thank you!