Functions which have harmonic support

Abstract

1. Introduction. This paper presents some properties of the functions in a certain subclass of the class of subharmonic functions. Consider real valued functions v(z) defined in a domain D of the complex plane. DEFINITION 1. The function v is in (hs), the class of functions which have harmonic support in D, if and only if v is uniformly bounded on each compact subset of D and for each zoC=Dah harmonic and single valued in D3h(zo) =v(zo) and h<v in D. Caratheodory [5] used such functions in constructing a general result on solutions of Dirichlet's problem which are based on convergent sequences of continuous operators. The functions of the class (hs) are continuous subharmonic functions, but Brelot [4, footnote 16] showed the existence of subharmonic functions in Cn which are not in (hs). A simple example of a continuous subharmonic function not in (hs) is the function v = z a, 0 <a <1, in zI <1. For suppose v C (hs) and let hn denote the support functions at Zn= l/n, n= 2, 3, * - - . The sequence { hn } constitutes a bounded family of harmonic functions which is thus normal. Writing z = x+iy, the family