Hyperbolic harmonic functions and the associated integral equations

Abstract

In this paper we consider a class of integral equations associated with the invariant mean value property of hyperbolic harmonic functions. We have shown that nonconstant solutions of these integral equations are functions of unbounded variations and do not attain their supremum or infimum on [0; 1): We present an algorithm to obtain numerical solutions of these integral equations. We also consider the equivalent ordinary differential equations and used MATLAB to obtain numerical solutions of these differential equations.

Keywords

• bergman space
• berezin transform
• hyperbolic harmonic functions
• invariant mean-value property
• integral equations