TITLE: On the upper bound of $\sum_{i=1}^{n}x^m_{i}$ subject to the conditions $\sum_{i=1}^{n}x_{i}=0$ and $\sum_{i=1}^{n}x^2_{i}=n$
QUESTION [5 upvotes]: The following question has been posted on mathematics stackexchange: inequalities problem, perhaps arising from a question on expectations.

Let $x_{1},x_{2},\cdots,x_{n}$ are real numbers, and such
$$\begin{cases}
x_{1}+x_{2}+\cdots+x_{n}=0\\
x^2_{1}+x^2_{2}+\cdots+x^2_{n}=n
\end{cases}$$
Let $\alpha_{m}=\displaystyle\dfrac{1}{n}\sum_{i=1}^{n}x^m_{i}$

See Mitrinovic D.S Analytic inequalities (Springer 1970) Page 347.
M.LAKSHMANAMURTI proved that
$$\alpha_{m}\le\dfrac{(n-1)^{m-1}+(-1)^m}{n(n-1)^{(m/2)-1}}.$$
I am interested in the details of the proof or a published reference.

REPLY [3 votes]: This is addressed in the following paper:
Rivin, Igor, Counting cycles and finite dimensional $L^{p}$ norms, Adv. Appl. Math. 29, No. 4, 647-662 (2002). ZBL1013.05042.