A note on harmonic functions

Document Type : Primary Research paper


Department of Mathematics, Maharani’s Science College for Women Mysuru, Karnataka, India – 570 005


Every function u regular and harmonic in a region G possesses for every sphere
S-centre P(x, yz) and radius R-lying entirely inside G the mean value property.
The converse of this thereon, that if U is continuous and possesses the property
(1) for every sphere is G then U is harmonic in G, was discovered by Bocher and
Koebe. In other words, the property (1) is a characteristic one for functions harmonic
in G. A number of conditions each of which characterises a harmonic function have
since been given for, instance Zaremba proved.