We prove the inequality h(x)−1 G(x, y)h(y) < cG(x, y) +c, where G is the Green function of a plane domain D, h is positive and harmonic on D, and c is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains c may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant c in the above inequality is finite.
ASJC Scopus subject areas
- Applied Mathematics