A Counterexample to a Conjecture by Błocki–Zwonek. Issue 1 (2nd January 2018)

• Record Type: Journal Article
• Authors: Åhag, P.; Czyż, R.; Lundow, P. H.
• Is Part Of: Experimental mathematics. Volume 27:Issue 1(2018)
• Page Start: 119
• Page End: 124
• Publication Date: 2018-01-02
• Subjects: Błocki–Zwonek conjectures -- Bergman kernel -- Green functions -- Jacobi functions; 32F45 -- 32U35 -- 32A25; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2016.1230913
• Languages: English
• ISSNs: 1058-6458

Abstract

ABSTRACT: For a bounded pseudoconvex domain and pluricomplex Green function g Ω ( z, a ) with pole at a ∈ Ω, it was conjectured by Błocki and Zwonek that β( t ) = log λ n ({ z ∈ Ω: g Ω ( z, a ) < t }) is a convex function on ( − ∞, 0). With Ω the annulus the Green function g Ω ( z, a ) with pole at a = 1 + 0i can be explicitly given in terms of Jacobi theta functions. We show numerically that in this case β is not convex.