Given a multinomial distribution $$ X = \left(X_{1},\ldots,X_{K}\right),\ \sum_{k = 1}^KX_{k} = n,\quad\mbox{and}\quad p_{1} = \cdots = p_{K} = \frac{1}{K}, $$ I wonder whether there is some approximation or lower bound (with respect to $K$, $n$, and $\alpha$) for the expected $\alpha$-power mean of $X$.
Specifically, I want to calculate the following: $$ \mathbb{E}_X\left[\left(% \sum_{k = 1}^{K}X_{k}^{\alpha}\right)^{1/\alpha} \,\,\right]\quad \mbox{given}\ \alpha\in \left(1,2\right). $$ I was checking the MGF of the multinomial distribution but I am not sure whether that is useful here. Any thoughts and ideas will be very helpful.