KöMaL 2016 B4775

Find those pairs $(n,k)$ of positive integers for which $\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}^{n}\ge \bigg(\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}\bigg)^{\! n}$ for all real numbers $a_1\ge a_2\ge \dots \ge a_{2k+1}\ge 0$.