Preface

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.