I assume you mean |m-X| as opposed to |m-EX|? Otherwise, |m-EX| is not a random variable, so E(|m-EX|^k) = |m-EX|^k is always zero (and hence minimized) when m = EX -- i.e., the mean -- and that's probably not what you're asking.
After a bit of Googling around, it looks like you might be talking about the third absolute central moment E(|X-EX|^3), which is related to something called the Barry-Esseen inequality ... see here.
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