TITLE: Possible to find a set of log-concave functions with log-concave sums?
QUESTION [5 upvotes]: While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is concave. This suggests one can find a set of log-concave functions whose sums are still log-concave (possibly closed under addition). 

I am interested in the concavity/convexity of $\ln(b+s)$, where $b,s$ are functions on $\mathbb{R}^+$, increasing, positive, convex and log-concave. 

Example: I found $\ln(x^2+x^\beta)''<0, \forall x>0$ for $\beta<6$  (numerical result, perhaps not exact). 

Are there any results that could help delineate this set of functions? Perhaps for polynomials?

Some suggestions and more detailed description here
The initial motivation for this problem comes from population dynamics.

REPLY [5 votes]: In fact $\ln(x^2 + x^\beta)$ is concave for $x > 0$ iff $3-2\sqrt{2} \le \beta \le 3 + 2 \sqrt{2}$. 
This comes from writing
$$ \dfrac{d^2}{dx^2} \ln(x^2 + x^\beta) = \dfrac{x^2}{(x^2 + x^\beta)^2} \left(-\beta x^{2\beta - 4} + (\beta^2 - 5 \beta + 2) x^{\beta - 2} -2\right)$$ 
and noting that the discriminant of $-\beta t^2 + (\beta^2 - 5 \beta + 2) t - 2$ with respect to $t$ is $(\beta^2 - 6 \beta + 1) (\beta - 2)^2$.
EDIT: Somewhat more generally, $\ln(x^\alpha + x^\beta)$ is concave for $x > 0$ iff
$\alpha=\beta$ or  $(\alpha-\beta)^2 - 2 (\alpha + \beta) + 1 \le 0$.