Let's say that we have function $u:\mathbb R_0\to \mathbb R$ with $u'(x)>0$, $u''(x)<0$, $u'''(x)>0$, $\lim_{x\to 0} u'(x) = \infty, \lim_{x\to 0}u'(x) = 0$.
Take $x_1 < x_2$. Does $$\frac{u'''(x_1)}{u'''(x_2)}\leq \frac{u'(x_1)}{u'(x_2)}$$ always hold?
Thanks, I'm lost here.