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An Application of the Schur Algorithm to Variability Regions of Certain Analytic Functions-I
Computational Methods and Function Theory ( IF 0.6 ) Pub Date : 2021-03-08 , DOI: 10.1007/s40315-021-00362-z
Md Firoz Ali , Vasudevarao Allu , Hiroshi Yanagihara

Let \(\Omega \) be a convex domain in the complex plane \({\mathbb C}\) with \(\Omega \not = {\mathbb C}\), and P be a conformal map of the unit disk \({\mathbb D}\) onto \(\Omega \). Let \({\mathcal F}_\Omega \) be the class of analytic functions g in \({\mathbb D}\) with \(g({\mathbb D}) \subset \Omega \). Also, let \(H_1^\infty ({\mathbb D})\) be the well known closed unit ball of the Banach space \(H^\infty ({\mathbb D})\) of bounded analytic functions \(\omega \) in \({\mathbb D}\), with norm \(\Vert \omega \Vert _\infty = \sup _{z \in {\mathbb D}} |\omega (z)|\). Let \({\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text {there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text {satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots ~\text {for} ~z\in \mathbb D\}\). For each fixed \(z_0 \in {\mathbb D}\), \(j=-1,0,1,2, \ldots \) and \(c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)\), we use the Schur algorithm to determine the region of variability \(V_\Omega ^j (z_0, c ) = \{ \int _0^{z_0} z^{j}(g(z)-g(0))\, d z : g \in {\mathcal F}_\Omega \; \text {with} \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}\). We also show that for \(z_0 \in {\mathbb D} \backslash \{ 0 \}\) and \(c \in \text {Int} \, {\mathcal C}(n) \), \(V_\Omega ^j (z_0, c )\) is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve \(\partial V_\Omega ^j (z_0, c )\).



中文翻译:

Schur算法在某些解析函数的可变区域中的应用-I

\(\ Omega \)是复平面\({\ mathbb C} \)中具有\(\ Omega \ not = {\ mathbb C} \)的凸域,而P是单位圆盘的共形映射\({\ mathbb D} \)\(\ Omega \)上。让\({\ mathcal F} _ \欧米茄\)是类的解析函数\({\ mathbb d} \)\(克({\ mathbb d})\子集\欧米茄\) 。另外,令\(H_1 ^ \ infty({\ mathbb D})\)是有界解析函数\ {的Banach空间\(H ^ \ infty({\ mathbb D})\)的众所周知的闭合单位球\欧米加\)\({\ mathbb d} \),使用范数\(\ Vert \ omega \ Vert _ \ infty = \ sup _ {z \ in {\ mathbb D}} | \ omega(z)| \)。令\({\ mathcal C}(n)= \ {(c_0,c_1,\ ldots,c_n)\ in {\ mathbb C} ^ {n + 1}:\ text {存在} \; \ omega \ in H_1 ^ \ infty({\ mathbb D})\; \ text {satisfying} \; \ omega(z)= c_0 + c_1z + \ cdots + c_n z ^ n + \ cdots〜\ text {for}〜z \ in \ mathbb D \} \)。对于每个固定的\(z_0 \ in {\ mathbb D} \)\(j = -1,0,1,2,\ ldots \)\(c =(c_0,c_1,\ ldots,c_n)\ in {\ mathcal C}(n)\),我们使用Schur算法确定变异性区域\(V_ \ Omega ^ j(z_0,c)= \ {\ int _0 ^ {z_0} z ^ {j}( g(z)-g(0))\,dz:g \ in {\数学F} _ \ Omega \; \ text {with} \;(P ^ {-1} \ circ g)(z)= c_0 + c_1z + \ cdots + c_n z ^ n + \ cdots \} \)。我们还显示了对于\(z_0 \ in {\ mathbb D} \反斜杠\ {0 \} \)\(c \ in \ text {Int} \,{\ mathcal C}(n)\)\( V_ \ Omega ^ j(z_0,c)\)是凸封闭的Jordan域,我们通过给出边界曲线\(\ partial V_ \ Omega ^ j(z_0,c)\)的参数表示来确定。

更新日期:2021-03-08
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