Remarks on the Grunsky norm and \(p\)th root transformation
type: Item
instance of: scholarly article
title: Remarks on the Grunsky norm and \(p\)th root transformation
zbMATH Open document ID: 0943.30013
DOI: 10.1016/S0377-0427(99)00017-5
author: Arcadii Z. Grinshpan
published in: Journal of Computational and Applied Mathematics
publication date: 2000-09-05T00:00:00Z
review text: Let \(S\) denote the class of functions \(f\) holomorphic and univalent in the unit disc \(E\), and normalized by the conditions \(f(0)= 0\), \(f'(0)=1\). Each function \(f\in S\) generates its Grunsky operator \(G_f= \{\sqrt{nm} \alpha_{n,m} \}_{n,m=1}^\infty: l^2\to l^2\), where \(\alpha_{n,m}\) are determined by the expansion \(\log [f(z)- f(\zeta)]/ (z-\zeta)= \sum_{n,m=0}^\infty \alpha_{n,m} z^n \zeta^m\), \(z,\zeta\in E\). Here \(l^2\) is the Hilbert space of all square-summable complex sequences \(X= (x_1, x_2,\dots)\), with the norm \(\|X\|= (\sum_{n=1}^\infty|x_n|^2)^{1/2}\). The author shows that the norm of the Grunsky operator generated by a univalent function does not decrease with the \(p\)th root transformation, i.e. there holds the theorem: Let \(f\in S\), \(p=2,3,\dots\), and let \(g\) be defined by the formula \(g(z)= \root p\of {f(z^p)}\), \(z\in E\). Then \(\|G_f\|\leq \|G_g\|\). This inequality is sharp for each \(p\) and over each subclass \(\{f\in S: \|G_f\|= k\}\), \(0\leq k\leq 1\).
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 1334872
zbMATH Keywords: univalent functions
zbMATH Keywords: Grunsky norm
zbMATH Keywords: \(p\)th root transformation
reviewed by: Zbigniew Jerzy Jakubowski
MaRDI profile type: Publication
cites work: Q4726525
cites work: Univalent functions with logarithmic restrictions
cites work: The Grunsky Norm and Some Coefficient Estimates for Bounded Functions
cites work: The Grunsky coefficient conditions
cites work: Q3331465
full work available at URL: s0377-0427(99)00017-5
OpenAlex ID: W2106318920
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description: scientific article; zbMATH DE number 1334872