An Application of the Schur Algorithm to Variability Regions of Certain Analytic Functions I
An Application of the Schur Algorithm to Variability Regions of Certain Analytic Functions I
复制标题
Schur算法在某些解析函数的可变区域中的应用I
DOI：
10.1007/s40315-021-00362-z
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发表时间：
2022
影响因子：
2.1
通讯作者：
Vasudevarao Allu and Hiroshi Yanagihara
中科院分区：
文献类型：
--
作者：
Md Firoz Ali;Vasudevarao Allu and Hiroshi Yanagihara
Let $$\Omega $$ Ω be a convex domain in the complex plane $${\mathbb C}$$ C with $$\Omega \not = {\mathbb C}$$ Ω ≠ C , andPbe a conformal map of the unit disk $${\mathbb D}$$ D onto $$\Omega $$ Ω . Let $${\mathcal F}_\Omega $$ F Ω be the class of analytic functionsgin $${\mathbb D}$$ D with $$g({\mathbb D}) \subset \Omega $$ g ( D ) ⊂ Ω . Also, let $$H_1^\infty ({\mathbb D})$$ H 1 ∞ ( D ) be the well known closed unit ball of the Banach space $$H^\infty ({\mathbb D})$$ H ∞ ( D ) of bounded analytic functions $$\omega $$ ω in $${\mathbb D}$$ D , with norm $$\Vert \omega \Vert _\infty = \sup _{z \in {\mathbb D}} |\omega (z)|$$ ‖ ω ‖ ∞ = sup z ∈ D | ω ( z ) | . Let $${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text {there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text {satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots ~\text {for} ~z\in \mathbb D\}$$ C ( n ) = { ( c 0 , c 1 , … , c n ) ∈ C n + 1 : there exists ω ∈ H 1 ∞ ( D ) satisfying ω ( z ) = c 0 + c 1 z + ⋯ + c n z n + ⋯ for z ∈ D } . For each fixed $$z_0 \in {\mathbb D}$$ z 0 ∈ D , $$j=-1,0,1,2, \ldots $$ j = - 1 , 0 , 1 , 2 , … and $$c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)$$ c = ( c 0 , c 1 , … , c n ) ∈ C …
Let $$\Omega $$ Ω be a convex domain in the complex plane $${\mathbb C}$$ C with $$\Omega \not = {\mathbb C}$$ Ω ≠ C , andPbe a conformal map of the unit disk $${\mathbb D}$$ D onto $$\Omega $$ Ω . Let $${\mathcal F}_\Omega $$ F Ω be the class of analytic functionsgin $${\mathbb D}$$ D with $$g({\mathbb D}) \subset \Omega $$ g ( D ) ⊂ Ω . Also, let $$H_1^\infty ({\mathbb D})$$ H 1 ∞ ( D ) be the well known closed unit ball of the Banach space $$H^\infty ({\mathbb D})$$ H ∞ ( D ) of bounded analytic functions $$\omega $$ ω in $${\mathbb D}$$ D , with norm $$\Vert \omega \Vert _\infty = \sup _{z \in {\mathbb D}} |\omega (z)|$$ ‖ ω ‖ ∞ = sup z ∈ D | ω ( z ) | . Let $${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text {there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text {satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots ~\text {for} ~z\in \mathbb D\}$$ C ( n ) = { ( c 0 , c 1 , … , c n ) ∈ C n + 1 : there exists ω ∈ H 1 ∞ ( D ) satisfying ω ( z ) = c 0 + c 1 z + ⋯ + c n z n + ⋯ for z ∈ D } . For each fixed $$z_0 \in {\mathbb D}$$ z 0 ∈ D , $$j=-1,0,1,2, \ldots $$ j = - 1 , 0 , 1 , 2 , … and $$c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)$$ c = ( c 0 , c 1 , … , c n ) ∈ C …