Subharmonic functions, generalizations, weighted boundary behavior, and separately subharmonic functions: A survey

Abstract

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 includes, among others, quasisubharmonic functions, nearly subharmonic functions (in a slightly generalized sense) and almost subharmonic functions (essentially). In addition to the basic properties of quasi-nearly subharmonic functions, we list certain weighted boundary behavior properties, and a counterpart to Armitage’s and Gardiner’s result concerning the subharmonicity of a separately subharmonic function. We slightly sharpen our previous improvement to Armitage’s and Gardiner’s result, too. In addition, we recall our recent improvements to Kołodziej’s and Thorbiörnson’s result concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second.