I'm looking at a paper by Arnold et al. (CPDE, 2001), in which they make use of convex functions in the context of relative entropies. There, they assume that on $(0,\infty)$, their entropy function satisfies $\psi(1)=0$, $\psi''>0$, and $(\psi''')^2 \leq \frac{1}{2}\psi'' \psi^{(4)}$ for all values in $(0,\infty)$. Form these assumptions, the authors are somehow able to infer that $\psi'''\leq 0$ in the set, a conclusion that totally escapes me. Since it's a minor aspect of a much larger paper, they don't give much attention to it. Can anyone clue me in?



Properties of Relative Entropies
If the assumption is $(\psi''')^2\le\frac12\,\psi''\psi^{(4)}$, then it can be written as $(1/\psi'')''\le0$. Therefore $1/\phi''$ is concave. Since it is positive over $(0,+\infty)$, it must be non-decreasing, that is $(1/\phi'')'\ge0$. This is precisely $\phi'''\le0$.


