Phragmén-Lindelöf's and Lindelöf's theorems
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.
scientific article; zbMATH DE number 3955333
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