Subharmonic functions, generalizations and separately subharmonic functions

Details Subharmonic functions, generalizations and separately subharmonic functions Subharmonic functions, generalizations and separately subharmonic functions

2006-10-08

arxiv.org/abs/math/0610259v5

Scientific Bulletin of Chelm, Section of Mathematics and Computer Science, No. 2/2007, pp. 49-76. (ISBN 978-83-61149-24-8)

Mathematics

Analysis of PDEs

A talk at the XIV Conference on Analytic Functions, July 22-28, 2007, Chelm, Poland. Some misprints of the previous version ar

Scientific paper

First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly subharmonic functions (in a slightly generalized sense) and almost subharmonic functions. We also give some basic properties of quasi-nearly subharmonic functions. Second, after recalling some of the existing subharmonicity results of separately subharmonic functions, we give the corresponding counterparts for separately quasi-nearly subharmonic functions, thus improving previous results of ours, of Lelong, of Avanissian and of Arsove. Third, we give two results concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second variable. The first result improves a result of Arsove, concerning the case when the function has, in addition, locally a negative integrable minorant. The second result improves a result of Ko{\l}odziej and Thorbi\"ornson concerning the subharmonicity of a function subharmonic and ${\mathcal{C}}^2$ in the first variable and harmonic in the second.

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