Subharmonic Functions, Generalizations, Holomorphic Functions, Meromorphic Functions, and Properties.

We give the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic functions (in a slightly generalized sense) and almost subharmonic functions (essentially). Moreover, we give basic properties of quasinearly subharmonic functions, and, among others, we characterize quasinearly subharmonicity with the aid of quasihyperbolic metric.

We find necessary and sufficient conditions under which subsets of balls are big enough for the characterization of nonnegative, quasinearly subharmonic functions by mean value inequalities. A similar result is obtained also for generalized mean value inequalities where, instead of balls, we consider arbitrary bounded sets which have nonvoid interiors and instead of the volume of ball we use functions depending on the radius of this ball.

Basing our proof on an old argument of Domar, we generalize and improve Armitage’s and Gardiner’s previous subharmonic function inequality result. Our result is stated for quasinearly subharmonic functions, it is rather general and, at the same time, flexible. Indeed, with the aid of it we will in the next two sections improve both Domar’s and our previous domination condition results of subharmonic functions, and also Armitage’s and Gardiner’s, and our results on the subharmonicity of separately subharmonic functions.

Using our version of a mean value type inequality for quasinearly subharmonic functions, presented in the previous section, we give domination conditions for families of quasinearly subharmonic functions. Our results improve previous results of Domar, Rippon and ours, and thus also the original results of Sjöberg and Brelot.

With the aid of our new version of a mean value type inequality, we improve our previous result of the subharmonicity of separately subharmonic functions, and thus also the well-known related earlier results of Armitage and Gardiner and ours. Moreover, we give refinements, with concise proofs, to the basic classical results of Avanissian, of Lelong, and of Arsove.

It is an open problem whether a function, subharmonic with respect to the first variable and harmonic with respect to the second, is subharmonic or not. Based again on our mean value type inequality, we improve our previous subharmonicity results of the above type functions, thus improving also the previous results of Kołodziej and Thorbiörnson and Imomkulov. Moreover, we give refinements, with concise proofs, to the older basic results of Arsove, and of Cegrell and Sadullaev.

We give certain weighted boundary behavior properties for quasinearly subharmonic functions, related to previous results of Gehring, Hallenbeck, Mizuta, Pavlovic, Stoll, Suzuki and others. Es- pecially, we give a limiting case result of a nonintegrability result of Suzuki.

It is an open problem whether a function, subharmonic with respect to the first variable and harmonic with respect to the second, is subharmonic or not. Based again on our mean value type inequality, we improve our previous subharmonicity results of the above type functions, thus improving also the previous results of Kołodziej and Thorbiörnson and Imomkulov. Moreover, we give refinements, with concise proofs, to the older basic results of Arsove, and of Cegrell and Sadullaev.

We give extension results for subharmonic, separately subharmonic, harmonic and separately harmonic functions. Our results improve previous results of Blanchet.

We give extension results for plurisubharmonic, convex and separately convex functions. Our results improve previous results of Blanchet.

Applying our extension result for subharmonic functions, we give an extension result for holomorphic functions, which is related to the well-known extension results of Besicovitch and Shiffman. In addition, we give related older extension results of holomorphic functions and of meromorphic functions.

We define locally uniformly homogeneous spaces, and consider quasinearly subharmonic functions there. Especially, we consider weighted boundary behavior and boundary integral inequalities of quasinearly subharmonic functions.