The behavior of functions harmonic and positive in an angular domain

The principal question to be investigated is the character of a function harmonic and positive in a simply connected domain G and zero on the boundary of G except possibly at a finite number of points. It is well known that if a function is harmonic and regular in a simply connected domain and zero everywhere on the boundary, then the function is identically zero. In fact, if the harmonic function is zero except possibly at a finite number of boundary points and if in a neighborhood of each of these points the function is bounded, then again the function is identically zero. Thus if a function is harmonic in G and zero on the boundary except at wo, a boundary point of G, and if the function is not identically zero, it must become infinite as w approaches wo. The purpose of this paper is to relate the growth of the function as w approaches wo to the character of the boundary of G in a neighborhood of wo. It is shown that the behavior or the growth of the function as w approaches wo depends on the size of the angular opening of G at wo. The precision with which the growth can be expressed depends on the regularity of the boundary of G in a neighborhood of wo