The study of exceptional sets for nontangential limits of harmonic functions has led to the problem of the equivalence of harmonic measure and surface measure on the boundary of Lipschitz domains. In this note we will review the development of these results. We will see how estimates of Green's function can be used to relate the measures and look at methods of obtaining the desired estimates. I...

It is shown that an analytic function taking circles to ellipses must be a Möbius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Möbius transformation. Analytic Möbius transformations take circles to circles. This is their most basic, most celebrated geometric property. We add the adjective ‘analytic’ because in a previous paper [1] we introduce...

THEOREM I. Let C be an arbitrary closed Jordan region of the (x, y)-plane, and let w = {z), z~x+iy, be a function which maps conformally the exterior of C onto the exterior of the unit circle in the w-plane so that the points at infinity correspond to each other. Let CR denote the curve \cf>(z) \=R, R>1, that is, the transform onto the z-plane of the circle \w\ = R. A necessary and sufficien...

In this paper we first establish several integral identities involving the multiple polylogarithm functions and Kaneko-Tsumura A-function, which can be thought as a single-variable function of level two. We find that these integrals expressed in terms zeta (star) values, their related variants (multiple t-values, T-values, S-values, etc.), harmonic sums T-harmonic sums, S-harmonic are closely t...