While reading papers about utility theory, I've stumbled upon a definition of a completely monotone function (AKA proper), which is a function with $u'>0$, $u''<0$, $u'''>0$ and so on. See http://mathworld.wolfram.com/CompletelyMonotonicFunction.html
The first conditions are straight forward: more is better than less and diminishing marginal utility. I'm not sure how to interpret higher-order conditions, and in general, what is so "complete" about this function?
Any intuitions are appreciated.
You ask for an intuition on the subject and this is one: $u'\gt0$ imposes that $u$ be increasing but does not give any restriction on the variation of $u'$. The additional condition $u''\lt0$ imposes on $u'$ the condition $u'$ decreasing . And so on. It is not an intrinsic property of the function $u$ but a problem to determine a function satisfying certain requirements.