The Bergman Analytic Content of Planar Domains

Abstract

Given a planar domain Ω, the Bergman analytic content measures the L 2 (Ω)-distance betweenz and the Bergman space A 2 (Ω). We compute the Bergman analytic content of simply-connected quadrature domains with quadrature formula supported at one point, and we also determine the function f ∈ A 2 (Ω) that best approximatesz. We show that, for simply-connected domains, the square of Bergman analytic content is equivalent to torsional rigidity from classical elasticity theory, while for multiply-connected domains these two domain constants are not equivalent in general.

Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions

“…Define the function R(a) : (0, ∞) → (0, ∞) by using the sum (5) with b = 1/a and letting j and k run from 0 to 85. It is clear that R(a) is an underestimate for ρ(Ω(a)), while we have already observed that ρ 12 (Ω(a)) is an overestimate for ρ(Ω(a)). Figure 1 shows a plot of ρ 12 (Ω(a))/R(a) − 1 for values of a between 0 and 10.…”

“…It is clear that R(a) is an underestimate for ρ(Ω(a)), while we have already observed that ρ 12 (Ω(a)) is an overestimate for ρ(Ω(a)). Figure 1 shows a plot of ρ 12 (Ω(a))/R(a) − 1 for values of a between 0 and 10. Notice that the error is smaller than one half of one percent.…”

“…The extremal problem defining Bergman analytic content is a minimization problem, so upper bounds can be obtained by trial and error. It was shown in [12] that the square of the Bergman analytic content of Ω is equal to ρ(Ω) when Ω is simply connected (as it is in the case we are considering), so in this case one can obtain both upper and lower bounds by trial and error.…”

Potential theory and approximation: highlights from the scientific work of Stephen Gardiner

“…Here, ρ(Ω) denotes the torsional rigidity of Ω and m is Lebesgue measure. Subsequently, Fleeman and Lundberg [12] showed that the left hand inequality in 14is actually an equality for any bounded simply connected domain, and this relationship has been further exploited by Fleeman and Simanek [13]. Bell,Ferguson and Lundberg [4] established related inequalities concerning torsional rigidity and the norm of the self-commutator of a Toeplitz operator.…”

Isoperimetric inequalities for Bergman analytic content

“…where ρ(Ω) denotes the torsional rigidity of Ω and m is Lebesgue measure. Subsequently, Fleeman and Lundberg [9] showed that the left hand inequality is actually an equality for any bounded simply connected domain, and this relationship has been further exploited by Fleeman and Simanek [10].…”