The greatest lower bound os the $n$th probability moment (1.1) of a population with variance $\sigma^2$ is given by (3.4).
"A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance." Ann. Math. Statist. 23 (2) 286 - 289, June, 1952. https://doi.org/10.1214/aoms/1177729447