Author: Juhani Riihentaus

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

eBook: US $69 Special Offer (PDF + Printed Copy): US $117
Printed Copy: US $83
Library License: US $276
ISBN: 978-981-14-9868-8 (Print)
ISBN: 978-981-14-9870-1 (Online)
Year of Publication: 2021
DOI: 10.2174/97898114987011210101


This book explains different types of subharmonic and harmonic functions. The book brings 12 chapters explaining general and specific types of subharmonic functions (eg. quasinearly subharmonic functions and other separate functions), related partial differential equations, mathematical proofs and extension results. The methods covered in the book also attempt to explain different mathematical analyses such as elliptical equations, domination conditions, weighted boundary behavior. The book serves as a reference work for scholars interested in potential theory and complex analysis.


Our presentation is divided into two parts. In the first part we consider subharmonic functions and their generalizations, so-called quasinearly subharmonic functions. In the second part we consider certain extension results for subharmonic functions, for holomorphic functions and for meromorphic functions.

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.

We are considering a rather general function class, namely quasinearly subharmonic functions. This 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 seems, nevertheless, to be justified. In some instances, the use of quasinearly subharmonic functions makes it possible to simplify and clarify certain proofs of subharmonic functions, and sometimes even improve the existing results. As examples, the subharmonicity results of separately subharmonic functions are presented in section 5, and the weighted boundary behavior results of subharmonic functions in section 7.

In the second part we consider removability results for subharmonic functions, for separately subharmonic functions, for harmonic functions, for separately harmonic functions, and for holomorphic and for meromorphic functions. Our results are related, at least slightly, to the well-known already existing results of holomorphic and meromorphic functions.


Not applicable.


The author declares no conflict of interest, financial or otherwise.


Declared none.

Juhani Riihentaus
Department of Mathematical Sciences
University of Oulu
P.O. Box 3000
FI-90014 Oulun yliopisto, Finland and
Department of Physics and Mathematics
University of Eastern Finland
P.O. Box 111
FI-80101 Joensuu


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