Reference for an estimate on the sum of powers $\sum_{k=1}^4 | x_k |^{2k}$ if $\sum_{k=1}^4 | x_k |^2 =1$?
Let $a_i = x_i^2\implies \displaystyle \sum_{i=1}^4 a_i = 1, 0 \le a_i \le 1\implies f(a_1,a_2,a_3,a_4) = a_1^k+a_2^k+a_3^k +a_4^k$.The max is $1$ when some of the $a_i = 1$ and the others are $0$. This is achieved by observing that the function is convex in each of the variables while holding the other three fixed.The min is $\dfrac{1}{4^{k-1}}$ by Jensen's inequality as the function $f(x) = x^k, k \ge 2$ is convex on $[0,1]$. If $k = 1$, the function is constant and is $1$.