Sharp and simple bounds for the raw moments of the binomial and Poisson distributions

We prove the inequality E[(X/μ)k]≤(k/μlog(1+k/μ))k≤exp(k2/(2μ)) for sub-Poissonian random variables X, such as Binomially or Poisson distributed variables, with mean μ. The asymptotic behaviour E[(X/μ)k]=1+O(k2/μ) matches a lower bound of 1+Ω(k2/μ) small k2/μ. This improves over previous uniform raw moment bounds by factor exponential in k.