On a problem of Turán concerning sums of powers of complex numbers
The paper deals with P. Turán's problem of evaluating \(R_ n= \min_{\mathbf z} \max_{1\leq j\leq n} | S_ j|\) under the condition \(1= {\mathbf z}_ 1\geq | {\mathbf z}_ 2|\geq \dots\geq | {\mathbf z}_ n|\), where \(S_ j= {\mathbf z}_ 1^ j+ \dots+ {\mathbf z}_ n^ j\). The best previously known result was due to F. V. Atkinson who proved that \(R_ n> S_ 0\) for a constant \(S_ 0\in (1/3, \pi/8)\). The author of this article proves that for every \(m\leq n\) and \({\mathbf z}\) such that \({\mathbf z}_ 1= {\mathbf z}_ 2= \dots= {\mathbf z}_ m\) we have \[ \max_{1\leq j\leq n-m+1} | S_ j|>m \Bigl( {\textstyle{{1\over 2} + {m\over {8n}}+ {{3m^ 2} \over {64n^ 2}}}} \Bigr). \] This result implies that \(R_ n> {1/2}\) and improves Atkinson's result.
scientific article; zbMATH DE number 638249