Geodesic
Schwarzian Derivative and Covering Arcs of a Pencil of Circles by Holomorphic Functions
Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 865-871.

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Let $f$ be a holomorphic function in the disk $U=\{z:|z|1\}$, $|f(z)|1$ in $U$, let $f(\pm1)=\pm1$ in the sense of angular limits, and let the angular Schwarzian derivatives $S_{f}(\pm1)$ exist. We establish an upper bound for the sum $S_{f}(-1)+S_{f}(1)$ under the assumption that the image $f(U)$ does not contain open arcs of the pencil of circles $\arg[(1+w)/(1-w)]=\theta$, $-\pi/2\theta\varphi$, with endpoints $w=\pm1$ and $$ \operatorname{Re} f''(1)+f'(1)(1-f'(1))=-\operatorname{Re} f''(-1)+f'(-1)(1-f'(-1))=0. $$ This bound depends on $\varphi$ and $f'(\pm1)$ only.
Keywords: Schwarzian derivative, holomorphic functions, boundary distortion, covering theorem. 
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     title = {Schwarzian {Derivative} and {Covering} {Arcs} of a {Pencil} of {Circles} by {Holomorphic} {Functions}},
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V. N. Dubinin. Schwarzian Derivative and Covering Arcs of a Pencil of Circles by Holomorphic Functions. Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 865-871. http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a6/

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