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• W2789807757 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a holomorphic and locally univalent function on the unit disk <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper D> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript r> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>C_r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the circle centered at the origin of radius <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=application/x-tex>r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than r greater-than 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>r</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0>r >1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We will prove that the total absolute curvature of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis upper C Subscript r Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(C_r)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an increasing function of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=application/x-tex>r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, we present inequalities involving the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper L Superscript p> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathrm {L}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm of the curvature of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis upper C Subscript r Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(C_r)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using the hyperbolic geometry of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper D> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we will prove an analogous monotonicity result for the hyperbolic total curvature. In the case where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a hyperbolically convex mapping of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper D> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into itself, we compare the hyperbolic total curvature of the curves <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript r> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>C_r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis upper C Subscript r Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(C_r)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and show that their ratio is a decreasing function. The last result can also be seen as a geometric version of the classical Schwarz Lemma." @default.
• W2789807757 created "2018-03-29" @default.
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• W2789807757 date "2018-03-05" @default.
• W2789807757 modified "2023-09-27" @default.
• W2789807757 title "Conformal mapping, convexity and total absolute curvature" @default.
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• W2789807757 doi "https://doi.org/10.1090/ecgd/317" @default.
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