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.