Sinks in Newton's means (Q5944887)

For \(n>1\), let \(F: \mathbb{C}^n \to \mathbb{C}^n\) be defined by \(F(x)=\widehat x\), where \(\widehat x=(\widehat {x}_1,\dots ,\widehat{x}_n)\) and \(\widehat {x}_k\) is the \(k\)th Newton's mean. The dynamical system \(\{F^N\}\) is discussed. Due to the Newton's inequalities, for \(x>0\), the iterates \(F^N(x)\to \sqrt{[n]} {\prod_{m=1}^n x_m}(1,\dots ,1)\). The convergence for complex \(x\) is investegated in this paper. It is proved that if \(H(x)\subset \mathbb{C}\backslash \{0\}\), where \(H(x)\) denotes the convex hall of \(x_1,\dots , x_n\in \mathbb{C}\), then also \(H(\widehat x)\subset \mathbb{C}\backslash \{0\}\) and \(F^N(x)\to x_0(1,\dots ,1)\) with \(x_0^n=\prod_{m=1}^n x_m\).