Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

1. Can an entire function be a (pretty) sigmoid?

If so, let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and composition.

2. Does $P$ contain a (pretty) sigmoid?