On extreme X-harmonic functions

A Class of Extreme X-harmonic Functions

Salisbury and Verzani introduced a class of martingales for the Brownian superprocess related to conditionings of the process to exit the boundary of a bounded domain in Rd in a particular way. The corresponding class of functions, denoted Hg,h1,...,hN , was generalized by Dynkin to more general superprocesses and shown to be X-harmonic. Salisbury and Verzani conjectured that a certain choice o...

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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