Notions of Convexity pp 116-224 | Cite as

# Subharmonic Functions

## Abstract

This chapter is in principle parallel to Chapter I: subharmonic functions in **R**^{n} are defined as convex functions on **R** with linear functions replaced by harmonic functions. However, this requires some background knowledge concerning harmonic functions which is provided in Section 3.1. Basic facts concerning subharmonic functions are then given in Section 3.2, mainly by means of the mean value property and the characterization as distributions with positive Laplacian. In Section 3.3 we start with the solution of the Dirichlet problem using Perron’s method before approaching the main topic, the Riesz representation formula, which generalizes Section 1.5. A number of applications to one variable analytic function theory are given. Finally, Section 3.4 is devoted to a study of the exceptional sets associated with subharmonic functions. It is proved that the polar sets where a subharmonic function can be equal to — ∞ are precisely the sets where the limit of a bounded increasing sequence of subharmonic functions can fail to be upper semi-continuous. This section will not be needed in the following chapters.

## Keywords

Harmonic Function Compact Subset Unit Disc Dirichlet Problem Positive Measure## Preview

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