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Parametric representation of functions in \(\Sigma\) and its application. (Chinese. English summary) Zbl 0633.30016

The Löwner differential equation for functions in \(\Sigma\), the set of functions \(F(\zeta)=\zeta +b_ 0+b_ 1/\zeta +..\). meromorphic and univalent in \(\{| \zeta | >1\}\), is derived from that for functions in S, the set of functions \(f(z)=z+..\). regular and univalent in \(\{| z| <1\}\). The \(\Sigma\)-Löwner chain, like the Löwner chain in S, is considered. \(\Sigma\)-Löwner chains of the form \[ f(\zeta,t)=e^{-t}\zeta +b_ m\zeta^{-m}+...,\quad m\geq 0, \] determine a set of functions \(f(\zeta)=f(\zeta,0)\) in \(\Sigma\) which is denoted by \(\Sigma_ m.\)
As an application the author proves the following theorem. Let \[ f(\zeta)=\zeta +b_ m\zeta^{-m}+...\in \Sigma \quad (m\geq 0). \] If either \(f\in \Sigma_ m\) or \((| \zeta |^ 2-1)| \zeta f''(\zeta)/f'(\zeta)| \leq 1\), then \[ | b_ n| \leq 2(n+1)^{-1},\quad m\leq n\leq 2m,\quad | b_ n| \leq 2(n+1)^{- 1}+4(n+1)^{-1}e^{-2(n+3)(n-1)},\quad n=2m+1. \] The estimates are sharp.
Reviewer: Liu Liquan

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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