2006/01/26 by Pascal J. Thomas, Thomas, Pascal J.
Mathematics · #32U35 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:32U35
paper · pdf · doi:10.48550/arxiv.math/0601642
16 pages; references added to related work of the author
openalex publication_date 2006/01/26 · arxiv created 2006/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Lempert function for several poles a0, ..., aN in a domain Ω of \mathbb Cn is defined at the point z ∈ Ω as the infimum of ∑Nj=0 log|ζj| over all the choices of points ζj in the unit disk so that one can find a holomorphic mapping from the disk to the domain Ω sending 0 to z. This is always larger than the pluricomplex Green function for the same set of poles, and in general different. Here we look at the asymptotic behavior of the Lempert function for three poles in the bidisk (the origin and one on each axis) as they all tend to the origin. The limit of the Lempert functions (if it exists) exhibits the following behavior: along all complex lines going through the origin, it decreases like (3/2) log |z|, except along three exceptional directions, where it decreases like 2 log |z|. The (possible) limit of the corresponding Green functions is not known, and this gives an upper bound for it.