Geodesic
Boundary values of the Schwarzian derivative of a~regular function
Sbornik. Mathematics, Tome 202 (2011) no. 5, pp. 649-663

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Regular functions $f$ in the half-plane $\operatorname{Im} z>0$ admitting an asymptotic expansion $f(z)=c_1z+c_2z^2+c_3z^3+\gamma(z)z^3$, where $c_1>0$, $\operatorname{Im} c_2=0$ and the angular limit $\angle\lim_{z\to0}\gamma(z)=0$, are considered. For various conditions on the function $f$ inequalities for the real part of the Schwarzian derivative $S_f(0)=6(c_3/c_1-c_2^2/c_1^2)$ are established. These inequalities complement and refine some known versions of Schwarz's lemma. The results obtained are close to the well-known Burns-Krantz rigidity theorem on regular self-maps and its generalizations due to Tauraso, Vlacci and Shoikhet. Bibliography: 16 titles.
Keywords: Schwarzian derivative, Schwarz's lemma, regular function. 
@article{SM_2011_202_5_a1,
     author = {V. N. Dubinin},
     title = {Boundary values of the {Schwarzian} derivative of a~regular function},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {202},
     number = {5},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2011_202_5_a1/}
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V. N. Dubinin. Boundary values of the Schwarzian derivative of a~regular function. Sbornik. Mathematics, Tome 202 (2011) no. 5, pp. 649-663. http://geodesic.mathdoc.fr/item/SM_2011_202_5_a1/