Let $B(n,x) = \sum_{k=0}^n {n\brace k}x^k$ be the Bell polynomials and $B_n = B(n,1)$ be the Bell numbers.

I recently proved a nice relation between the two: $$ B(n,x)^{1/n}/x \ge B_{n/x}^{x/n}, $$ where we used the generalization of the Bell numbers to real indicies: $B_z = \frac{1}{e}\sum_{k=0}^\infty \frac{k^z}{k!}$. I wonder if this is a known result?

I also conjecture a matching upper bound $$ B(n,x)^{1/n}/x \le B_{n/x+1}^{1/(n/x+1)}. $$

I have checked this numerically, but not been able to prove it. I tried using an exponential-generating function bound on $B(n,x)$, but that appears to not be strong enough.

I wonder if this is a known result as well? I read the article Berend and Tassa, but there doesn't appear to be anything of this kind.

Alternatively, am I missing a simple proof?

Since $B(n,x)$ is also the $n$th raw moment of a Poisson variable with mean $x$, such a result would tie those moments in a simple, uniform way to the Bell numbers.