Boundary values of the Schwarzian derivative of a regular function

V. N. Dubinin

Sbornik. Mathematics, Tome 202 (2011) no. 5, pp. 649-663

Résumé

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.

Détail

Keywords: Schwarzian derivative, Schwarz's lemma, regular function.

Bibliographie

[1] V. N. Dubinin, “Schwarz's lemma and estimates of coefficients for regular functions with free domain of definition”, Sb. Math., 196:11 (2005), 1605–1625 | DOI | MR | Zbl

[2] R. Osserman, “A sharp Schwarz inequality on the boundary”, Proc. Amer. Math. Soc., 128:12 (2000), 3513–3517 | DOI | MR | Zbl

[3] V. N. Dubinin, “The Schwarz inequality on the boundary for functions regular in the disk”, J. Math. Sci. (N. Y.), 122:6 (2004), 3623–3629 | DOI | MR | Zbl

[4] H. P. Boas, “Julius and Julia: mastering the art of the Schwarz lemma”, Amer. Math. Monthly, 117:9 (2010), 770–785 | DOI | MR | Zbl

[5] S. G. Krantz, “The Schwarz lemma at the boundary”, Complex Var. Elliptic Equ. (to appear)

[6] D. M. Burns, S. G. Krantz, “Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary”, J. Amer. Math.Soc., 7:3 (1994), 661–676 | DOI | MR | Zbl

[7] R. Tauraso, F. Vlacci, “Rigidity at the boundary for holomorphic self-maps of the unit disk”, Complex Variables Theory Appl., 45:2 (2001), 151–165 | DOI | MR | Zbl

[8] D. Shoikhet, “Another look at the Burns–Krantz theorem”, J. Anal. Math., 105:1 (2008), 19–42 | DOI | MR | Zbl

[9] M. D. Contreras, S. Diaz-Madrigal, Ch. Pommerenke, “Second angular derivatives and parabolic iteration in the unit disk”, Trans. Amer. Math. Soc., 362:1 (2010), 357–388 | DOI | MR | Zbl

[10] P. S. Bourdon, J. H. Shapiro, “Cyclic phenomena for composition operators”, Mem. Amer. Math. Soc., 125, no. 4, Amer. Math. Soc., Providence, RI, 1997 | MR | Zbl

[11] V. N. Dubinin, “Symmetrization in the geometric theory of functions of a complex variable”, Russian Math. Surveys, 49:1 (1994), 1–79 | DOI | MR | Zbl

[12] I. P. Mityuk, “The symmetrization principle for multiply connected domains”, Soviet Math. Dokl., 5:1 (1964), 928–930 | MR | Zbl

[13] I. P. Mityuk, Simmetrizatsionnye metody i ikh primenenie v geometricheskoi teorii funktsii. Vvedenie v simmetrizatsionnye metody, Kubanskii gos. un-t, Krasnodar, 1980

[14] V. N. Dubinin, “Majorization principles for meromorphic functions”, Math. Notes, 84:5–6 (2008), 751–755 | DOI | MR

[15] S. Stoilow, Teoria functiilor de o variabila complexa, v. 2, Editura Academiei Republicii Populare Romine, Bucuresti, 1958 | MR | Zbl

[16] W. K. Hayman, Multivalent functions, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl