Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Multi-point variations of the Schwarz lemma with diameter and width conditions

Author: Dimitrios Betsakos
Journal: Proc. Amer. Math. Soc. 139 (2011), 4041-4052
MSC (2010): Primary 30C80
Published electronically: March 28, 2011
MathSciNet review: 2823049
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ f$ is holomorphic in the unit disk $ \mathbb{D}$ and $ f(\mathbb{D})\subset \mathbb{D}$, $ f(0)=0$. A classical inequality due to Littlewood generalizes the Schwarz lemma and asserts that for every $ w\in f(\mathbb{D})$, we have $ \vert w\vert\leq \prod_j \vert z_j(w)\vert$, where $ z_j(w)$ is the sequence of pre-images of $ w$. We prove a similar inequality by replacing the assumption $ f(\mathbb{D})\subset \mathbb{D}$ with the weaker assumption Diam $ f(\mathbb{D})=2$. This inequality generalizes a growth bound involving only one pre-image, proven recently by Burckel et al. We also prove growth bounds for holomorphic $ f$ mapping $ \mathbb{D}$ onto a region having fixed horizontal width. We give a complete characterization of the equality cases. The main tools in the proofs are the Green function and the Steiner symmetrization.

References [Enhancements On Off] (What's this?)

  • 1. H.P. Boas, Julius and Julia: Mastering the art of the Schwarz lemma. Amer. Math. Monthly 117 (2010), 770-785.
  • 2. A. Baernstein II, Integral means, univalent functions and circular symmetrization. Acta Math. 133 (1974), 139-169. MR 0417406 (54:5456)
  • 3. D. Betsakos, Equality cases in the symmetrization inequalities for Brownian transition functions and Dirichlet heat kernels. Ann. Acad. Sci. Fenn. Math. 33 (2008), 413-427. MR 2431373 (2009e:60175)
  • 4. D. Betsakos, Geometric versions of Schwarz's lemma for quasiregular mappings. Proc. Amer. Math. Soc. 139 (2011), 1397-1407.
  • 5. A.F. Beardon, D. Minda, A multi-point Schwarz-Pick lemma. J. Anal. Math. 92 (2004), 81-104. MR 2072742 (2005f:30044)
  • 6. R.B. Burckel, D.E. Marshall, D. Minda, P. Poggi-Corradini, T.J. Ransford, Area, capacity and diameter versions of Schwarz's lemma. Conform. Geom. Dyn. 12 (2008), 133-152. MR 2434356 (2010j:30050)
  • 7. V.N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable. Russian Math. Surveys 49 (1994), 1-79. MR 1307130 (96b:30054)
  • 8. W.K. Hayman, Multivalent Functions. Second edition. Cambridge University Press, 1994. MR 1310776 (96f:30003)
  • 9. W.K. Hayman, Subharmonic Functions, Vol. 2. Academic Press, 1989. MR 1049148 (91f:31001)
  • 10. O. Lehto, On the distribution of values of meromorphic functions of bounded characteristic. Acta Math. 91 (1954), 87-112. MR 0062226 (15:947c)
  • 11. J.E. Littlewood, Lectures on the Theory of Functions. Oxford University Press, 1944. MR 0012121 (6:261f)
  • 12. R. Nevanlinna, Analytic Functions. Springer, 1970. MR 0279280 (43:5003)
  • 13. R. Osserman, From Schwarz to Pick to Ahlfors and beyond. Notices Amer. Math. Soc. 46 (1999), 868-873. MR 1704258 (2000i:30049)
  • 14. G. Pólya, G. Szegö, Problems and Theorems in Analysis. I. Springer, 1978. MR 580154 (81e:00002)
  • 15. S. Pouliasis, Equality cases for condenser capacity inequalities under symmetrization. Preprint, 2010.
  • 16. T. Ransford, Potential Theory in the Complex Plane. Cambridge University Press, 1995. MR 1334766 (96e:31001)
  • 17. R. Remmert, Theory of Complex Functions. Springer-Verlag, 1991. MR 1084167 (91m:30001)
  • 18. J.H. Shapiro, The essential norm of a composition operator. Ann. of Math. (2) 125 (1987), 375-404. MR 881273 (88c:47058)
  • 19. V. A. Shlyk, A uniqueness theorem for the symmetrization of arbitrary capacitors. Siberian Math. J. 23 (1982), 267-287. MR 0652233 (84i:31002)
  • 20. M. Vuorinen, Conformal Geometry and Quasiregular Mappings. Lecture Notes in Math., 1319, Springer-Verlag, 1988. MR 950174 (89k:30021)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30C80

Retrieve articles in all journals with MSC (2010): 30C80

Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Keywords: Holomorphic function, Schwarz lemma, Steiner symmetrization, capacity, Green function, inner function, diameter, width.
Received by editor(s): September 30, 2010
Published electronically: March 28, 2011
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society