Approximation of subharmonic functions in the unit disk

TL;DR

This paper demonstrates that subharmonic functions in the unit disk can be approximated by the logarithm of analytic functions with bounded mean difference, and also explores uniform approximation methods.

Contribution

It establishes the existence of an analytic function approximating subharmonic functions in the disk with bounded integral difference and discusses uniform approximation techniques.

Findings

Existence of an analytic function f with bounded integral difference from u

Bounded constant C independent of u

Discussion of uniform approximation methods

Abstract

Let u be a subharmonic function in D={|z|<1}. There exist an absolute constant C and an analytic function f in D such that \int_D |u(z)-log|f(z)|| dm(z)<C where m denotes the plane Lebesgue measure. We also consider uniform approximation.