Bohr's radius for polynomials in one complex variable
It is known that if \(f(z)=\sum^\infty_{k=0}a_kz^k\) is analytic in \(\mathbb{D}=\{z:|z| <1\}\) and \(\|f\|_\infty=\sup_{|z|<1}<1\), then \(\sum^\infty_{k=0}|a_k|r^k\leq 1\) for all \(r\leq\frac 13\) and the constant \(1/3\), often called the Bohr radius, cannot be improved. Let \(\Pi_n= \{p_n(z)=a_0+a_1z+\cdots+a_nz^n\), \(a_n\neq 0\), \(\|p_n\|= 1\}\). We call \(R_n\) the Bohr radius for \(\Pi_n\), if \(\sum^n_{k=0}|a_k|r^k \leq 1\) for all \(r\leq R_k\) and all \(p_n\in\Pi_n\), and if \(R_n\) is the largest such number. In this paper the following estimates are derived: If \(R_n\) is the Bohr radius for \(\Pi_n\), then there exist the constants \(C_1\), \(C_2\) with \(0< C_1\), \(C_2<\infty\) such that \[ C_1\cdot 3^{-n/2}<R_n-\tfrac 13<C_2\tfrac 1n\log n \] for all \(n\in\mathbb{N}\). For \(n\) tending to infinity, we can see that \[ R_\infty =\lim_{n\to\infty}R_n=\tfrac 13. \]
scientific article; zbMATH DE number 5011341