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Domination conditions for families of quasinearly subharmonic functions. (English) Zbl 1220.31007
Summary: Y. Domar [Ark. Mat. 3, 429–440 (1958; Zbl 0078.09301)] has given a condition that ensures the existence of the largest subharmonic minorant of a given function. Later, P. J. Rippon [Math. Scand. 49, 128–132 (1981; Zbl 0472.31002)] pointed out that a modification of Domar’s argument gives in fact a better result. Using our previous, rather general and flexible, modification of Domar’s original argument, we extend their results both to the subharmonic and quasinearly subharmonic settings.
MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31C45 Other generalizations (nonlinear potential theory, etc.)
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References:
[1] Y. Domar, “On the existence of a largest subharmonic minorant of a given function,” Arkiv för Matematik, vol. 3, no. 39, pp. 429-440, 1957. · Zbl 0078.09301 · doi:10.1007/BF02589497
[2] N. Sjöberg, “Sur les minorantes sousharmoniques d’une fonction donnée,” in Neuvième Congrès des mathematiciens Scandinaves, pp. 309-319, Helsingfors, 1938. · JFM 65.0421.01
[3] M. Brelot, “Minorantes sous-harmoniques, extrémales et capacités,” Journal de Mathématiques Pures et Appliquées. Neuvième Série, vol. 24, pp. 1-32, 1945. · Zbl 0061.22802
[4] J. W. Green, “Approximately subharmonic functions,” Proceedings of the American Mathematical Society, vol. 3, pp. 829-833, 1952. · Zbl 0048.34103 · doi:10.2307/2032187
[5] P. J. Rippon, “Some remarks on largest subharmonic minorants,” Mathematica Scandinavica, vol. 49, no. 1, pp. 128-132, 1981. · Zbl 0472.31002 · eudml:166763
[6] Y. Domar, “Uniform boundedness in families related to subharmonic functions,” Journal of the London Mathematical Society. Second Series, vol. 38, no. 3, pp. 485-491, 1988. · Zbl 0631.31002
[7] M. Hervé, Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces, Lecture Notes in Mathematics, Vol. 198, Springer, Berlin, Germany, 1971. · Zbl 0214.36404
[8] J. Riihentaus, “On a theorem of Avanissian-Arsove,” Expositiones Mathematicae, vol. 7, no. 1, pp. 69-72, 1989. · Zbl 0677.31004
[9] J. Riihentaus, “Subharmonic functions, generalizations and separately subharmonic functions,” in XIV Conference on Analytic Functions, vol. 2 of Scientific Bulletin of Chełm, Section of Mathematics and Computer Science, pp. 49-76, Chełm, Poland, 2007. · Zbl 1137.31302
[10] M. Pavlović, “On subharmonic behaviour and oscillation of functions on balls in \Bbb Rn,” Institut Mathématique. Publications. Nouvelle Série, vol. 55(69), pp. 18-22, 1994. · Zbl 0824.31003 · eudml:118716
[11] M. Pavlović and J. Riihentaus, “Classes of quasi-nearly subharmonic functions,” Potential Analysis, vol. 29, no. 1, pp. 89-104, 2008. · Zbl 1158.31002 · doi:10.1007/s11118-008-9089-1
[12] J. Riihentaus, “Subharmonic functions, generalizations, weighted boundary behavior, and separately subharmonic functions: A survey,” in 5th World Congress of Nonlinear Analysts (WCNA ’08), Orlando, Fla, USA, 2008. · Zbl 1239.31002
[13] J. Riihentaus, “Subharmonic functions, generalizations, weighted boundary behavior, and separately subharmonic functions: A survey,” Nonlinear Analysis, Series A: Theory, Methods & Applications, vol. 71, no. 12, pp. e2613-e26267, 2009. · Zbl 1239.31002 · doi:10.1016/j.na.2009.05.077
[14] D. H. Armitage and S. J. Gardiner, “Conditions for separately subharmonic functions to be subharmonic,” Potential Analysis, vol. 2, no. 3, pp. 255-261, 1993. · Zbl 0788.31005 · doi:10.1007/BF01048509
[15] J. Riihentaus, “Quasi-nearly subharmonicity and separately quasi-nearly subharmonic functions,” Journal of Inequalities and Applications, vol. 2008, Article ID 149712, 15 pages, 2008. · Zbl 1152.31006 · doi:10.1155/2008/149712 · eudml:129473
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