An application of the Schur algorithm to variability regions of certain analytic functions

2019/05/24 by Ali, Md Firoz, Allu, Vasudevarao, Yanagihara, Hiroshi
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.10241

Abstract

Let Ω be a convex domain in the complex plane \mathbb C with Ω\not= \mathbb C, and P be a conformal map of the unit disk \mathbb D onto Ω. Let \mathcal F_Ω be the class of analytic functions g in \mathbb D with g(\mathbb D) ⊂ Ω, and H₁^∞ (\mathbb D) be the closed unit ball of the Banach space H^∞ (\mathbb D) of bounded analytic functions ω in \mathbb D, with norm ‖ ω‖_∞ = sup_z ∈ \mathbb D |ω(z)|. Let \mathcal C(n) = \ (c₀,c₁ , … , c_n ) ∈ \mathbb Cⁿ⁺¹: there exists ω∈ H₁^∞ (\mathbb D) satisfying ω(z) = c₀+c₁z + ⋯ + c_n zⁿ + ⋯ for z∈ \mathbb D\. For each fixed z₀ ∈ \mathbb D, j=-1,0,1,2, … and c = (c₀, c₁ , … , c_n) ∈ \mathcal C(n), we use the Schur algorithm to determine the region of variability V_Ω^j (z₀, c ) = \ ∫₀^z₀ z^j(g(z)-g(0)) d z : g ∈ \mathcal F_Ω with (P^-1 ∘ g) (z) = c₀ +c₁z + ⋯ + c_n zⁿ + ⋯ \. We also show that for z₀ ∈ \mathbb D \backslash \ 0 \ and c ∈ \textrmInt \mathcal C(n) , V_Ω^j (z₀, c ) is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve ∂ V_Ω^j (z₀, c ).

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