Far Eastern Mathematical Journal

To content of the issue


Two-point boundary distortion estimate for Schwarzian derivative of holomorphic function


V. N. Dubinin

2014, issue 2, Ñ. 191–199


Abstract
Let $f$ be a holomorphic function in the disk $|z|<1$, $|f(z)|<1$, and let $z_1, z_2$ are distinct boundary points of this disk in which the angular limits $f(z_k), k=1,2$, exist, $f(z_{1})\neq f(z_{2})$, $|f(z_1)|=|f(z_2)|=1$. Under some geometric constraints on f the precise upper bound for $\Re\{S_{f}(z_{1})+S_{f}(z_{2})\}$ is established. Here $S_{f}(z)$ means the Schwarzian derivative of the function $f$ at the point $z$.

Keywords:
Schwarzian derivative, holomorphic functions, boundary distortion

Download the article (PDF-file)

References

[1] S. Kim, D. Minda, “Two-point distortion theorems for univalent function”, Pacific J. Math., 163 (1994), 137–157.
[2] W. Ma, D. Minda, “Two-point distortion for univalent functions”, J. Comput. Appl. Math., 105 (1999), 385–392.
[3] J. A. Jenkins, “On two-point distortion theorems for bounded univalent regular functions”, Kodai Math. J., 24:3 (2001), 329–338.
[4] D. Kraus, O. Roth, “Weighted distortion in conformal mapping in euclidean, hyperbolic and elliptic geometry”, Ann. Acad. Sci. Fenn. Math., 31 (2006), 111–130.
[5] M. D. Contreras, S. D\'{i}az-Madrigal, A. Vasil'ev, “Digons and angular derivatives of analytic self-maps of the unit disk”, Complex Variables and Elliptic Equations, 52:8 (2007), 685–691.
[6] J. M. Anderson, A. Vasil'ev, “Lower Schwarz–Pick estimates and angular derivatives”, Ann. Acad. Sci. Fenn., 33 (2008), 101–110.
[7] V. Bolotnikov, M. Elin, D. Shoikhet, “Inequalities for angular derivatives and boundary interpolation”, Anal. Math. Phys., 3:1 (2013), 63–96.
[8] A. Frolova, M. Levenshtein, D. Shoikhet, A. Vasil'ev, “Boundary distortion estimates for holomorphic maps”, Published online: 18 December 2013, Complex Anal. Oper. Theory.
[9] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer, 1992.
[10] V. N. Dubinin, V. Yu. Kim, “Teoremy iskazheniya dlya regulyarnyx i ogranichennyx v kruge funkcij”, Zap. nauchn. semin. POMI, 350, 2007, 26–39.
[11] 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.
[12] D. Shoikhet, “Another look at the Burns–Krantz theorem”, J. Anal. Math., 105:1 (2008), 19–42.
[13] V. N. Dubinin, “O granichnyx znacheniyax proizvodnoj Shvarca regulyarnoj funkcii”, Matem. sb., 202:5 (2011), 29–44.
[14] G. M. Goluzin, Geometricheskaya teoriya funkcij kompleksnogo peremennogo, Nauka, M., 1966.
[15] V. N. Dubinin, Emkosti kondensatorov i simmetrizaciya v geometricheskoj teorii funkcij kompleksnogo peremennogo, Dal'nauka, Vladivostok, 2009.
[16] W. K. Hayman, Myltivalent functions, Cambridge Univ. Press., Cambridge, 1994.
[17] V. N. Dubinin, “Lemma Shvarca i ocenki koe'fficientov dlya regulyarnyx funkcij so svobodnoj oblast'yu opredeleniya”, Matem. sb., 196:11 (2005), 53–74.

To content of the issue