On some power sum problems of Turán and Erdös
The author studies some problems connected with P. Turán's power sum problem \(\#10\) of finding \(\inf \max_{1\leq v\leq n^2} |\sum^n_{k=1} z^v_k |\), where the infimum is taken over all \(z_j\in \mathbb{C}\) satisfying \(|z_j |\geq 1\). He proves Theorem 1. If \(z_k\) are complex numbers and \(1\leq m\leq n\), then \[ \max_{v=1, \dots, 2mn- m(m+ 1)+1} \Biggl|\sum^n_{k=1} z^v_k \Biggr|\Biggl/ \sqrt {\sum^m_{k=1} |z_k |^{2v}} \geq 1. \] Some corollaries of the Theorem are obtained: Corollary 1. If \(1\leq m\leq n\) and \(|z_k |\geq 1\), then \[ \max_{v=1, \dots, 2mn- m(m+ 1)+1} \Biggl|\sum^n_{k=1} z^v_k \Biggr|\geq \sqrt {m}. \] Corollary 2. If \(|z_k |\geq 1\), then \[ \max_{v=1, \dots, 2n-1} \Biggl|\sum^n_{k=1} z^v_k \Biggr|\geq 1\qquad \qquad (\text{a result of Cassels}). \] Corollary 3. If \(|z_k |\geq 1\), then \[ \max_{v=1, \dots, n^2- n+1} \Biggl|\sum^n_{k=1} z^v_k \Biggr|\geq \sqrt {n}. \]
scientific article; zbMATH DE number 878136