Let $N$ be a poisson variable with mean $\lambda$. I seek proposals of a "pretty" increasing positive function $\psi(x)$ such that $\mathbb{E}\psi^{-1}(N)^p<\infty$ for some $p>1$ but $\mathbb{E}\psi^{-1}(N)^{p+1}=\infty.$ Here $\psi^{-1}$ denotes the generalized inverse. The smaller $p$ the better.
By pretty i mean:
- $\psi$ is algebraic, i.e. it has a closed form. Simpler is better.
- If that is not possible, a "known" named function $\psi$ is also appreciated.
Details: The MGF of the poisson variable $t\mapsto \mathbb{E}\exp(tN)$ exists for all $t\in \mathbb{R_+}$ so $\psi$ needs to be asymptotically slower than the logarithm. The inequalities at section 4.5 in this wiki article suggests that $\psi^{-1}$ should asymptotically look like $xe^x$, but the inverse to this function has no closed form.