EDIT: I am interested in derivative of a function $f(x)$. However, since the function is logarithmically convex/concave it is easier to analyze $\log f(x)$. Therefore, I rewrote the derivative and analyze the following product: $$h(x)=f(x)g(x),$$ where $g(x) = \frac{d}{dx}\log f(x)$, i.e., $$h(x)=f(x)(\log(f(x))^\prime.$$
I know that function $f(x)$ is positive and strictly increasing ($f^\prime(x)>0$) and furthermore logarithmically convex for $x<0$ and logarithmically concave for $x>0$, i.e., $g(x)>0$ and $\frac{d}{dx}g(x)=0$ for $x=0$. Also, $g(x)$ is an even function.
Thus, I know that there is exactly one maximum of $g(x)$.
I would like to show that also the function $h(x)$ has exactly one maximum. Intuitively, since I multiply the function $g(x)$ which has exactly one maximum and is even with a function $f(x)$ which is increasing (and moreover, related to $g(x)$), the maximum can only be shifted but no other maxima can be created or disappear.
My approach was to take the derivative of $h(x)$ and look if I can find some $x_0$ such that for $x<x_0$ $h(x)^\prime > 0$ and for $x>x_0$, $h(x)^\prime<0$, i.e., $$h^\prime(x)=f^\prime(x)(\log f(x))^\prime+f(x)(\log f(x))^{\prime\prime}=0.$$
Clearly, the first part is always positive. The second part is positive for $x<0$ and negative for $x>0.$ My problem is that if I add these two things up, it can possibly become a mess with more roots, or can't it?
Am I missing some more properties which would guarantee this? Any hints or ideas how to proceed? Thanks.
$$ h(x)=f(x)\frac{d}{dx}\,(\log f(x))=f(x)\,\frac{f'(x)}{f(x)}=f'(x). $$ Since $f$ is logarithmically convex on $(-\infty,0)$ it is also convex. This means that $f'$ is increasing on $(-\infty,0)$. Similarly, $f'$ is decreasing on $(0,\infty)$. This implies that $f'$ attains a maximum at $x=0$. However $f'$ could be constant on an interval around $x=0$. This would not happen $f$ were strictly logarithmically convex on $(-\infty,0)$ and strictly logarithmically concave on $(0,\infty)$
Note: the part in bold is false in general. Concavity implies log-concavity, but not the other way around.