Acta Mathematica Vietnamica
Study on Special Features of New h-functions of Some Simply Connected Planar Regions
Arunmaran Mahenthiram
,
Piriyalucksan Paramesvarampillai
Abstract
The $h$-function $h(r)$, or the harmonic-measure distribution function, is defined as the probability that a Brownian particle released from a basepoint $z_0$ in a region ${\varOmega }$ first leaves the region ${\varOmega }$ within distance $r$ of $z_0$. In this paper, we compute the $h$-function formulas explicitly in terms of $r$ for some new simply connected planar regions by using the method of conformal maps. Especially, we investigate the behaviour of these $h$-functions at certain values of $r$ that correspond to specific points on the boundary of the respective regions ${\varOmega }$. In some cases, we explain the asymptotic behaviour as r decreases to the minimum distance between the basepoint $z_0$ and the boundary.
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