Stronger distortion theorems of univalent functions and its applications
type: Item
instance of: scholarly article
title: Stronger distortion theorems of univalent functions and its applications
zbMATH Open document ID: 0538.30013
published in: Chinese Annals of Mathematics. Series B
publication date: 1983-01-01T00:00:00Z
review text: In this paper, by means of the representation theorem of continuous linear functionals on the space of continuous functions, the author improves and generalizes the famous FitzGerald inequalities, the Bazilevich inequalities and the Hayman regular theorem. For example: Theorem III.1. Suppose \(f\in S\), \(\{\gamma_{k,n}\}\) are the Grunsky coefficients of \(g(s)=1/f(z)=\zeta +b_ 1\zeta^{-1}+b_ 2\zeta^{- 2}+...,\quad z=1/\zeta,\quad \theta_ 0=\theta_ 0(f)\) and \(\alpha_ f\) are the Hayman direction and Hayman constant of f respectively, \(\ln(f(z)/z)=2(\gamma_ 1z+\gamma_ 2z^ 2+...).\) If \(\sum^{\infty}_{k=1}| \eta_ k|^ 2<\infty,\) then we have the sharp inequality \[ (frac{1}{2}\ln 1/\alpha_ f- \sum^{\infty}_{n=1}n| \gamma_ n-(1/n)\exp(-in \theta_ 0)|^ 2) \] \[ \times(\sum^{\infty}_{k=1}(1/k)| \eta_ k|^ 2-\sum^{\infty}_{n=1}n| \sum^{\infty}_{k=1}\eta_ k\gamma_{k,n}|^ 2) \] \[ \geq | \sum^{\infty}_{k=1}\eta_ k[((1/k)\exp(-ik \theta_ 0)-\gamma_ k)+\sum^{\infty}_{n=1}n\overline{(\gamma_ n-(1/n)\exp(-in \theta_ 0))}\gamma_ n|^ 2. \] Theorem V.1. Suppose \(F(\zeta)=\zeta +b_ 1\zeta^{-1}+b_ 2\zeta^{-2}+...\in \Sigma ',\quad G(w)=F^{- 1}(w)=w+B_ 1w^{-1}+B_ 2w^{-2}+...,\) and \(\{\lambda_{p,q}\}\) are the Grunsky coefficients. Then we have \[ | \sum^{\infty}_{m,n=1}\lambda_{m,n}\chi_ m\chi_ n| \leq \sum^{\infty}_{n=1}(1/n)| \chi_ n+\sum^{\infty}_{m=n+2}B_ m^{(-n)}\chi_ m|^ 2, \] where \(B_ m^{(-n)}\) is defined as \([G(w)]^{-k}=w^{-k}+\sum^{\infty}_{n=k+2}B_ n^{(-k)}w^{- n}.\)
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 3855438
zbMATH Keywords: FitzGerald inequalities
zbMATH Keywords: Bazilevich inequalities
zbMATH Keywords: Hayman regular theorem
zbMATH Keywords: Grunsky coefficients
author: Fuyao Ren
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3855438