Upper bound for the $t$-th moment in terms of lower moments

Question

Let $a_1, \ldots, a_n$ be positive integers. For positive integers $t$ and $m$ define the sum $$ M_t(m) = \dfrac{1}{n} \sum_{k=1}^n |a_k - m|^t. $$ I'm interested in upper bounds for $M_t(m)$ in terms of lower moments $M_s(m)$ with $s < t$. Specifically I'm interested in upper-bounding moments of the type $M_{4t+2}(m)$ in terms of $M_{4t}(m), M_{4}(m), M_2(m), M_1(m)$. Would you know or could you derive a good upper bound in terms of these?