Let $f(t)$ be a monic real polynomial such that $f(t) > 0$ for all $t \ge 0$. Suppose that $\log f(e^x)$ is strictly convex on $\mathbb{R}$, i.e. $f(s^2) \cdot f(t^2) > f(st)^2$ for all $s, t \ge 0$. Can one show that in fact $$\frac{d^2}{dx^2}(\log f(e^x)) > 0$$ for all $x \in \mathbb{R}$?
Remark: Since $\log f(e^x)$ is strictly convex (and hence convex), we always have $\frac{d^2}{dx^2}(\log f(e^x)) \ge 0$.
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