On positive harmonic functions in the upper half space

In the n(≧2)-dimensional Euclidean (ξ)=(ξ_1.…, ξ_n) space, let v (ξ) be a positive harmonic function in the upper half space G : ξ_n>0 and H be the boundary of G, that is, the hyperplane ξ_n=0. Then, denoting by η=(η_1,…, η_, 0) the points of the plane H, there exists a non-negative mass distribution υin H and a constant c≧0 such that [numerical formula] where [numerical formula] denotes the surface area of the unit sphere [numerical formula] In this note, we shall present, in §3,a new proof of the above formula (*), which is entirely different from the original proof and seems to be more simple and more natural as compared with the original one. Moreover, as its application, we shall give, in §5,a extremely brief proof for the classical but epoch-making Fatou's therem in the theory of analytic functions and, by means of the real function-theoretic arguments, we shall give, in §6,a delicate variant of the Reflection Princiole due to H. Schwarz. The first two sections 1 and 2 give the prelimiaries for the following treatments.departmental bulletin pape