Sharp and Simple Bounds for the raw Moments of the Binomial and Poisson Distributions

We prove the inequality $E[(X/\mu)^k] \le (\frac{k/\mu}{\log(k/\mu+1)})^k \le \exp(k^2/(2\mu))$ for sub-Poissonian random variables, such as Binomially or Poisson distributed random variables with mean $\mu$. The asymptotics $1+O(k^2/\mu)$ can be shown to be tight for small $k$. This improves over previous uniform bounds for the raw moments of those distributions by a factor exponential in $k$.