SCHWEITZER 1976 Problem 5

Let $S_{\nu}=\sum_{j=1}^n b_jz_j^{\nu} \;(\nu=0,\pm 1, \pm 2 ,...)$ , where the $b_j$ are arbitrary and the $z_j$ are nonzero complex numbers . Prove that $$|S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.$$

G. Halasz