Phragmén-Lindelöf's and Lindelöf's theorems
type: Item
instance of: scholarly article
title: Phragmén-Lindelöf's and Lindelöf's theorems
zbMATH Open document ID: 0594.30022
DOI: 10.1007/BF02384420
published in: Arkiv för Matematik
publication date: 1985-01-01T00:00:00Z
review text: Prior to the paper under review Phragmén-Lindelöf- and Lindelöf- type theorems in the present setup were proved by several mathematicians as the authors point out. Earlier results were obtained by \textit{V. M. Miklyukov} [Mat. Sb., Nov. Ser. 111(153), 42-66 (1980; Zbl 0428.35036)], \textit{V. A. Botvinnik} and \textit{V. M. Miklyukov} [Sib. Mat. Zh. 21, No.2, 232-235 (1980; Zbl 0435.46027)], \textit{S. Rickman} and \textit{M. Vuorinen} [Ann. Acad. Sci. Fenn. Ser. A I 7, 221-231 (1982; Zbl 0516.30018)], \textit{M. Vuorinen} [(*) Complex analysis, Banach Cent. Publ. 11, 353-362 (1983; Zbl 0543.30015)] and \textit{S. Granlund} [Manuscr. Math. 36, 355-365 (1981; Zbl 0506.35033)] as well as by the present authors [Ann. Acad. Sci. Fenn., Ser. A I 7, 233-247 (1982; Zbl 0468.30015)]. Continuing their series of investigations (for a nice survey of their earlier joint work see Martio's survey ''Nonlinear potential theory'', pp. 65-104 in Summer School in Potential Theory ed. by \textit{I. Laine} and \textit{O. Martio}, Joensuu Korkeakoulu Joensuu 1983), the authors prove several interesting theorems. They make systematic use of the notion of F-harmonic measure and the fact (due to Yu. G. Reshetnyak) that if \(f: G\to R^ n\) is a quasiregular mapping, then log\(| f|\) satisfies in \(G\setminus f^{-1}(0)\) a nonlinear elliptic PDE. The authors also make use of \(\sup er\)-f\({}^{\#}F\)-extremals, which they introduce, and succeed in proving natural generalizations of some classical results in the present context. It would require some additional effort to compare their Lindelöf-type theorems involving F-harmonic densities to those of the reviewer (*) involving capacity densities even in the special case of quasiconformal mappings.
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 3955333
zbMATH Keywords: Phragmén-Lindelöf theorem
zbMATH Keywords: Lindelöf-type theorems
zbMATH Keywords: F-harmonic measure
zbMATH Keywords: quasiregular mapping
author: Seppo Granlund
author: Peter Lindqvist
author: O. Martio
reviewed by: M. Vuorinen
MaRDI profile type: Publication
cites work: Q5179675
cites work: Rings and Quasiconformal Mappings in Space
cites work: Conformally Invariant Variational Integrals
cites work: F-harmonic measure in space
cites work: Note on the PWB-method in the nonlinear case
cites work: Estimates of harmonic measures
cites work: Q3103241
cites work: Q4139197
cites work: Reflection principle for solutions of elliptic partial differential equations and quasiregular mappings
cites work: Q5683049
cites work: Q5583267
cites work: Q5518064
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Q4405418
cites work: Asymptotic values and angular limits of quasiregular mappings of a ball
cites work: Q3259806
cites work: On the boundary behavior of locally K-quasiconformal mappings in space
cites work: On the existence of angular limits of \(n\)-dimensional quasiconformal mappings
cites work: Capacity Densities and Angular Limits of Quasiregular Mappings
description: scientific article; zbMATH DE number 3955333