Harmonic functions play a crucial role in mathematics. The same is true for a generalized class, for subharmonic functions. In this important area many authors, to mention just a few, Szpilrajn, Radó, Brelot, Lelong, Avanissian, Hervé, and Lieb and Loss, have found it useful to consider more general function classes, namely quasisubharmonic functions, nearly subharmonic functions, and almost subharmonic functions.
The main topic of the book 'Subharmonic Functions, Generalizations, Holomorphic Functions, Meromorphic Functions, and Properties' is quasinearly subharmonic functions. This new, quite recently defined class includes quasisubharmonic functions, nearly subharmonic functions and even almost subharmonic functions, at least more or less. Our class has its roots at least in the late fifties. The class of quasinearly subharmonic functions includes, in addition to nearly subharmonic functions, also functions satisfying certain growth conditions, especially certain eigenfunctions, polyharmonic functions, and subsolutions of certain general elliptic equations. Since harmonic functions are included in our class, nonnegative solutions of some elliptic equations are included. In particular, the partial differential equations associated with quasiregular mappings belong to this family of elliptic equations. Though the class of quasinearly subharmonic functions is indeed large, the use of it is justified. With the aid of quasinearly subharmonic functions we are able to simplify and clarify certain proofs of subharmonic functions, and in certain cases to improve the existing results. Examples, among others, are subharmonicity results of separately subharmonic functions, domination condition results for families of subharmonic functions and weighted boundary behavior results of subharmonic functions. Moreover, we give removability results for subharmonic functions, for separately subharmonic functions, for harmonic functions, for separately harmonic functions, and for holomorphic and for meromorphic functions.
About the Author:
Juhani Riihentaus was born in 1942, in Helsinki, Finland. He has completed his Ph.D. in Mathematics in 1975 from the University of Helsinki. He was a Docent at the University of Eastern Finland and at the University of Oulu (1980-2010). He has also worked as a research fellow at Mittag-Leffler Institute (1976-1977 and 1987). He has also served as a visiting researcher at the Ukrainian Academy of Sciences, Kiev (1995) and at Wuhan Institute of Physics and Mathematics, China (1995). His research areas include mathematics, potential theory and complex analysis of several complex variables. He has published 53 papers in several international journals.
Subharmonic, nearly subharmonic, quasinearly subharmonic, plurisubharmonic, convex, holomorphic, meromorphic functions, quasihyperbolic metric, families of quasinearly subharmonic functions, domination conditions, separately quasinearly subharmonic functions, extension, boundary behavior, approach regions, radial order, Ahlfors-regular sets, Hausdorff measure, net measure, Minkowski content, n-small set, locally uniformly homogeneous space.
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