The Wikipedia page on risk aversion states that a "Constant Relative Risk Aversion implies a Decreasing Absolute Risk Aversion, but the reverse is not always true". Let me decompose this statement in two parts:
1/ "Constant Relative Risk Aversion implies a Decreasing Absolute Risk Aversion."
A simple example is the log utility function, $u(c) = \ln(c)$, with $c>0$ satisfies the DARA because the utility function is positively skewed $\left(u'''=\frac{2}{c^3} >0\right)$ and implies a Relative Risk Aversion equals to $1 \left(=-c\frac {u''(c)}{u'(c)}\right)$.
2/ "but the reverse is not always true".
I am wondering if this is the most frequent case? Or if most of the time DARA utility functions also exhibit CRRA?
I would be grateful if you can illustrate your answer with some utility functions.