2000/04/28 by Eric Schippers · 4 citations
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Differential Equations and Boundary Problems
paper · pdf · doi:10.1090/s0002-9939-00-05623-9
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex"> \mathcal S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the class of functions which are univalent and holomorphic on the unit disc. We derive a simple differential equation for the Loewner flow of the Schwarzian derivative of a given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of script upper S overTilde"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f ∈ \mathcal S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This is used to prove bounds on higher order Schwarzian derivatives which are sharp for the Koebe function. As well we prove some two-point distortion theorems for the higher order Schwarzians in terms of the hyperbolic metric.