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• W2089257625 abstract "A basic theorem of iteration theory (Henrici [6]) states that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$ id=E1><mml:mi>f</mml:mi></mml:math>analytic on the interior of the closed unit disk<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$D$ id=E2><mml:mi>D</mml:mi></mml:math>and continuous on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$D$ id=E3><mml:mi>D</mml:mi></mml:math>with<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{Int}}left( D right)$ id=E4><mml:mtext>Int</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$fleft( D right)$ id=E5><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>carries any point<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$zvarepsilon D$ id=E6><mml:mi>z</mml:mi><mml:mi> ϵ </mml:mi><mml:mi>D</mml:mi></mml:math>to the unique fixed point<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$alpha varepsilon D$ id=E7><mml:mi>α</mml:mi><mml:mi> ϵ </mml:mi><mml:mi>D</mml:mi></mml:math>of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$ id=E8><mml:mi>f</mml:mi></mml:math>. That is to say,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f^n left( z right) to alpha $ id=E9><mml:msup><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>→</mml:mo><mml:mi>α</mml:mi></mml:math>as<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$n to infty $ id=E10><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:math>. In [3] and [5] the author generalized this result in the following way: Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$F_n left( z right): = f_1 circ ldots circ f_n left( z right)$ id=E11><mml:msub><mml:mi>F</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>∘</mml:mo><mml:mo>…</mml:mo><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. Then<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f_n to f$ id=E12><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mi>f</mml:mi></mml:math>uniformly on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$D$ id=E13><mml:mi>D</mml:mi></mml:math>implies<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$F_n left( z right)lambda $ id=E14><mml:msub><mml:mi>F</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>λ</mml:mi></mml:math>, a constant, for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$zvarepsilon D$ id=E15><mml:mi>z</mml:mi><mml:mi> ϵ </mml:mi><mml:mi>D</mml:mi></mml:math>. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structures<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f_1 circ ldots circ f_n left( z right)$ id=E16><mml:msub><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>∘</mml:mo><mml:mo>…</mml:mo><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>where the<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f_j $ id=E17><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>'s are analytic on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{Int}}left( D right)$ id=E18><mml:mtext>Int</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>and continuous on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$D$ id=E19><mml:mi>D</mml:mi></mml:math>with<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{Int}}left( D right)$ id=E20><mml:mtext>Int</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f_j left( D right)$ id=E21><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, but essentially random. Applications include analytic functions defined by this process." @default.
• W2089257625 created "2016-06-24" @default.
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• W2089257625 date "1991-01-01" @default.
• W2089257625 modified "2023-09-24" @default.
• W2089257625 title "Inner composition of analytic mappings on the unit disk" @default.
• W2089257625 doi "https://doi.org/10.1155/s0161171291000236" @default.
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