Modified Poisson Kernel and Integral Representation of Harmonic Functions in Half-Plane

Abstract:

      In this paper, using a property of the modified Poisson kernel in a half plane, we prove that a harmonic function $u(z)$ in a half plane with its positive part $u^{+}(z)=\max\{u(z),0\}$ satisfying a slowly growing condition can be represented by its integral in the boundary of the half plane and that its negative part $u^{-}(z)=\max\{-u(z),0\}$ can be dominated by a similar slowly growing condition. This improves some classical results about harmonic functions in a half-plane.