I am currently doing research involving power series on the unit disk in $\Bbb C$: precisely I am studying the properties of converging power series of a standard form $$ f(z)= \sum_{n=0}^\infty a_n z^n \label{1}\tag{1} $$ where (assuming conventionally for the discourse that $0\notin\Bbb N$)

  1. $\limsup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}=1$, and
  2. $z\in \Bbb D=\{z\in\Bbb C : |z|<1\}$.

From the basic properties of $\limsup$, condition 1 implies the boundedness of the set $\big\{|a_n|^{\frac 1n}\big\}_{n\in\Bbb N}$ , i.e. it implies the existence of $\sup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}$, and this has some important consequences on the structure of \eqref{1}, namely an estimate of the size of zero free regions, the size of the minimal univalence radius etc., described in an old paper by Milos Kössler [1].

The question

The paper [1] focuses on what a finite value of $\sup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}$ implies: does there exist other studies trying to analyze the implications of a given behavior for $n \mapsto |a_n|^{\frac 1n}$, $n\in \Bbb N$ or for the related $n \mapsto \sup_{k>n}|a_k|^{\frac 1k}$?


[1] Milos Kössler, "O významu čísla $\sup |a_n|^{\frac 1n}$ v teorii mocninných řad (The signification of the number $\sup |a_n|^{\frac 1n}$ in the theory of power series)", (Czech, English summary), Časopis Pro Pěstování Matematiky a Fysiky 74, No. 1, 47-53 (1949), MR0034833, Zbl 0033.26504


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