I'm studying relationships between trace entropy functionals and combinatorics and I'm faced with the following problem. Lets $\mathcal {D}$ be the following differential operator $1 -x\cdot \cfrac{d}{dx}$ i.e. $\mathcal {D} g = g - x\cdot g'$.


For $m\ge 0$ integer, if $\Phi\_m(x) := x\cdot \log(x)^m$ then $\mathcal {D} \Phi\_m(x) = -m\cdot \Phi\_{m-1}(x)$ and, (at least formally) for a function $g$ we can write
$g(x) = \sum\limits\_{m\ge 0} g\_m \cdot \Phi\_m(x)$ where $$g\_m = \cfrac{(-1)^m}{m!}\cdot \mathcal {D}^m g(x) |\_{x=1}\;.$$


I am trying to figure out what kind of functions $g(x)$ can satisfy the following conditions on $(0;1]$


I) $g(x) \ge 0$, $g(0)=0$. (Positivity)


II) $\mathcal {D}^2 g(x) \le 0$. (some sort of $\mathcal {D}$-concavity)


III) $\left(\mathcal {D}^2 - \mathcal {D}\right) g(x) \le 0$ (this is standard concavity expressed with the operator $\mathcal {D}$


IV) $\exists \ 0< \varepsilon \le 1$ such that $g(x) > -x^2\cdot \log(x) \quad \forall \ x \in (0;\varepsilon)$


V) $\exists \ a \in (0;\frac{1}{2}]$ such that $$g(a)+g(1-a) = -a\cdot \log(a) -(1-a)\cdot \log(1-a).$$


If we impose only conditions I),II) and III) there are a lot of functions satisfying them, but


1. adding IV) I'm not able to find any function except of the following form $g(x) = k\cdot x \cdot \log(x), \ k$ real constant (here $\varepsilon=1$). Note that $g$ do not satisfy V).
2. adding IV) and V) I'm not able to find any other function except the Boltzman-Gibbs-Shannon entropy trace : $-x\cdot \log(x)$


I "fear" that the Boltzman-Gibbs-Shannon entropy trace is the unique function satisfying I)-V).


Thanks in advance for any point of view.



Trace entropies
For any $c\in(0,\log2]$, the function $g$ defined by the formula $g(x)=cx$ for $x\in[0,1]$ satisfies your conditions I)–V), but it is not the Boltzman–Gibbs–Shannon entropy trace.




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There are many more functions $g$ satisfying your conditions I)–V) that are not the Boltzman–Gibbs–Shannon entropy trace. In particular, for any $c\_1\in(0,\log2)$, any nonnegative continuous function $H$ on $[0,1]$, and any small enough real $c\_2\ge0$ and $c\_3\ge0$, the function $g$ defined by the formula
$$g(x)=c\_1x-c\_2 x\log x-c\_3 x\int\_0^x du\,H(u)\log\frac xu$$
for $x\in[0,1]$ satisfies your conditions I)–V), but it is not the
Boltzmann–Gibbs–Shannon entropy trace.


