Nice bounds for 3rd folded central moment in terms of variance?

Question

Say we have some real numbers $x_1, \ldots, x_n$, and let $\mu$ be their mean and $\sigma^2$ be the variance. Are there some nice bounds for $\frac 1n \sum_{i=1}^n |x_i - \mu|^3$ based on $\sigma^2$? Maybe as a starter we could use that for $x,y\geq 0$, $(x+y)^{3/2} \geq x^{3/2} + y^{3/2}$ (though I don't know how to prove this off the top of my head -- if someone has a short proof that'd be nice to put in the comments) and so $$\frac 1n \sum_{i=1}^n |x_i - \mu|^3 = \frac 1n \sum_{i=1}^n (|x_i - \mu|^2)^{3/2} \leq \frac 1n \left(\sum_{i=1}^n |x_i-\mu|^2\right)^{3/2} = \frac 1n n^{3/2} \sigma^3 = \sigma^3\sqrt n$$ Are there better bounds?