On α-Harmonic Functions

Estimates and structure of α-harmonic functions

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open sets D. This yields a unique representation of such functions as integrals against measures on Dc∪{∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test. 1 Main results and ...

More on Harmonic Functions

On Univalent Harmonic Functions

Two classes of univalent harmonic functions on unit disc satisfying the conditions ∑∞ n=2(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) and ∑∞ n=2 n(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) are given. That the ranges of the functions belonging to these two classes are starlike and convex, respectively. Sharp coefficient relations and distortion theorems are given for these functions. Furthermore results concerning th...

On Harmonic Functions on Trees

By a tree T we mean a connected graph such that every subgraph obtained from T by removing any of its edges is not connected. In what follows we will only consider trees in which we distinguish a vertex v0 as an origin. As usual we denote by V and E the set of vertices and the set of edges (respectively) of the tree. If v and w are the boundary vertices of an edge, we say that they are neighbou...

Preprint LECTURES ON HARMONIC FUNCTIONS

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