Let X be a Poisson variable with parameter λ with a probability mass function, f(k), where k = 0, 1, 2 … We know the index of log-concavity is the function rf(k) = f(k)^2/(f(k-1)f(k+1) = (k+1)/k>1. So, Poisson is log-concave.
Many books have: "The random variable Y is dispersive if, and only if, Y has a logconcave density."
Poisson variable is discrete. Do we have: A Poisson variable X is dispersive?