Let $S$ be the smallest set of functions from $\mathbb{R} \mapsto \mathbb{R}$ such that

$$\forall n \in \mathbb{N}, (\alpha, \beta) \in \mathbb{R}^2, (f, g, h) \in S^2 \implies (\alpha f + \beta e^g + h)^n \in S$$

And $x \mapsto x \in S$

i.e. $S$ contains the identity and is closed under linear combination, exponentiation and raising to an integer power.

Does $S$ contain a function $f$ such that

$$\forall x \in \mathbb{R},\\0<f(x)<1\\\wedge~f(x)+f(-x)=1\\\wedge~f'(x) > 0\\\wedge~f''(x) (f(0)-f(x)) \ge 0$$

i.e. a sigmoid?

$S$ is of interest because for $f$ in $S$ the space $T(f) = \{x \mapsto f(x-\alpha)^n, n \in \mathbb{N},\alpha \in \mathbb{R}\}$ can easily be parametrized with a finite number of parameters. There are applications in computational statistics if such a function exists.