Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Two-point distortion theorems for harmonic and pluriharmonic mappings


Authors: Peter Duren, Hidetaka Hamada and Gabriela Kohr
Journal: Trans. Amer. Math. Soc. 363 (2011), 6197-6218
MSC (2010): Primary 32H02; Secondary 30C45
DOI: https://doi.org/10.1090/S0002-9947-2011-05596-0
Published electronically: July 26, 2011
MathSciNet review: 2833550
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Two-point distortion theorems are obtained for affine and linearly invariant families of harmonic mappings on the unit disk, with generalizations to pluriharmonic mappings of the unit ball in $ {\mathbb{C}}^{n}$. In particular, necessary and sufficient conditions are given for a locally univalent harmonic or pluriharmonic mapping to be univalent. Some particular subclasses are also considered.


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

  • 1. C. Blatter, Ein Verzerrungssatz für schlichte Funktionen, Comment. Math. Helv. 53 (1978), 651-659. MR 511855 (80d:30010)
  • 2. M. Chuaqui, P. Duren, and B. Osgood, Two-point distortion theorems for harmonic mappings, Illinois J. Math. 53 (2009), 1061-1075. MR 2741178
  • 3. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I 9 (1984), 3-25. MR 752388 (85i:30014)
  • 4. P. L. Duren, Univalent Functions, Springer-Verlag, New York, 1983. MR 708494 (85j:30034)
  • 5. P. Duren, Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, U. K., 2004. MR 2048384 (2005d:31001)
  • 6. P. Duren and R. Weir, The pseudohyperbolic metric and Bergman spaces in the unit ball, Trans. Amer. Math. Soc. 359 (2007), 63-76. MR 2247882 (2007k:32005)
  • 7. K. Fan, Distortion of univalent functions, J. Math. Anal. Appl. 66 (1978), 626-631. MR 517751 (80a:30015)
  • 8. S. Gong, Convex and Starlike Mappings in Several Complex Variables, Kluwer Acad. Publ., Dordrecht, 1998. MR 1689825 (2000c:32054)
  • 9. I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc., New York, 2003. MR 2017933 (2004i:32002)
  • 10. I. Graham, G. Kohr and J. Pfaltzgraff, Growth and two-point distortion for biholomorphic mappings of the ball, Complex Var. Elliptic Equ. 52 (2007), 211-223. MR 2297771 (2008a:32016)
  • 11. L. Hörmander, On a theorem of Grace, Math. Scand. 2 (1954), 55-64. MR 0062844 (16:27b)
  • 12. M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter & Co., Berlin-New York, 1993. MR 1242120 (94k:32039)
  • 13. J. A. Jenkins, On weighted distortion in conformal mapping $ II$, Bull. London Math. Soc. 30 (1998), 151-158. MR 1489326 (98m:30038)
  • 14. J. A. Jenkins, On two point distortion theorems for bounded univalent regular functions, Kodai Math. J. 24 (2001), 329-338. MR 1866369 (2002h:30012)
  • 15. S. A. Kim and D. Minda, Two-point distortion theorems for univalent functions, Pacific J. Math. 163 (1994), 137-157. MR 1256180 (94m:30042)
  • 16. G. Kohr, On some distortion results for convex mappings in $ \mathbb{C}^{n}$, Complex Variables Theory Appl. 39 (1999), 161-175. MR 1717651 (2000f:32018)
  • 17. S. G. Krantz, Function Theory of Several Complex Variables, Reprint of the 1992 Edition, AMS Chelsea Publishing, Providence, RI, 2001. MR 1846625 (2002e:32001)
  • 18. D. Kraus and O. Roth, Weighted distortion in conformal mapping in Euclidean, hyperbolic and elliptic geometry, Ann. Acad. Sci. Fenn. Math. 31 (2006), 111-130. MR 2210112 (2007j:30013)
  • 19. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692. MR 1563404
  • 20. W. Ma and D. Minda, Two-point distortion theorems for bounded univalent functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), 425-444. MR 1469801 (98i:30011)
  • 21. W. Ma and D. Minda, Two-point distortion theorems for strongly close-to-convex functions, Complex Variables Theory Appl. 33 (1997), 185-205. MR 1624935 (99b:30019)
  • 22. W. Ma and D. Minda, Two-point distortion for univalent functions, J. Comput. Appl. Math. 105 (1999), 385-392. MR 1690605 (2000e:30029)
  • 23. A. W. Marshall, I. Olkin and F. Proschan, Monotonicity of ratios of means and other applications of majorization, In: Inequalities (O. Shisha, editor), Academic Press, New York, 1967, pp. 177-190. MR 0237727 (38:6008)
  • 24. P. T. Mocanu, Sufficient conditions of univalency for complex functions in the class $ C^{1}$, Anal. Numer. Theor. Approx. 10 (1981), 75-79. MR 670636 (84a:30025)
  • 25. J. A. Pfaltzgraff, Distortion of locally biholomorphic maps of the $ n$-ball, Complex Variables Theory Appl. 33 (1997), 239-253. MR 1624947 (99a:32030)
  • 26. J. A. Pfaltzgraff, Koebe transforms of holomorphic and harmonic mappings, Talk presented at conference in Lexington, Kentucky, May 2008.
  • 27. J. A. Pfaltzgraff and T. J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie-Sklodowska Sect. A 53 (1999), 193-207. MR 1778828 (2001h:32023)
  • 28. J. A. Pfaltzgraff and T. J. Suffridge, Linear invariance, order and convex maps in $ \mathbb{C}^{n}$, Complex Variables Theory Appl., vol. 40, 1999, pp. 35-50. MR 1742869 (2000i:32026)
  • 29. J. A. Pfaltzgraff and T. J. Suffridge, Norm order and geometric properties of holomorphic mappings in $ {\mathbf{C}}^{n}$, J. Analyse Math. 82 (2000), 285-313. MR 1799667 (2001k:32028)
  • 30. Ch. Pommerenke, Linear-invariante Familien analytischer Funktionen $ I$, Math. Ann. 155 (1964), 108-154. MR 1513275
  • 31. Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
  • 32. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
  • 33. K. A. Roper and T. J. Suffridge, Convex mappings on the unit ball of $ {\mathbf{C}}^{n}$, J. Analyse Math. 65 (1995), 333-347. MR 1335379 (96m:32023)
  • 34. O. Roth, A distortion theorem for bounded univalent functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), 257-272. MR 1921307 (2003j:30040)
  • 35. O. Roth, Distortion theorems for bounded univalent functions, Analysis 23 (2003), 347-369. MR 2052374 (2005a:30024)
  • 36. W. Rudin, Function Theory in the Unit Ball of $ \mathbb{C}^{n}$, Springer-Verlag, New York, 1980. MR 601594 (82i:32002)
  • 37. T. Sheil-Small, Constants for planar harmonic mappings, J. London Math. Soc. 42 (1990), 237-248. MR 1083443 (91k:30052)
  • 38. T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes in Math. Vol. 599, Springer-Verlag, 1977, 146-159. MR 0450601 (56:8894)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32H02, 30C45

Retrieve articles in all journals with MSC (2010): 32H02, 30C45


Additional Information

Peter Duren
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
Email: duren@umich.edu

Hidetaka Hamada
Affiliation: Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku Fukuoka 813-8503, Japan
Email: h.hamada@ip.kyusan-u.ac.jp

Gabriela Kohr
Affiliation: Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogăl- niceanu Str., 400084 Cluj-Napoca, Romania
Email: gkohr@math.ubbcluj.ro

DOI: https://doi.org/10.1090/S0002-9947-2011-05596-0
Keywords: Harmonic mapping, pluriharmonic mapping, two-point distortion, affine invariance, linear invariance, univalence, convex mapping, close-to-convex mapping, starlike mapping
Received by editor(s): August 7, 2009
Published electronically: July 26, 2011
Additional Notes: The second author was partially supported by Grant-in-Aid for Scientific Research (C) No. 22540213 from Japan Society for the Promotion of Science, 2011.
The third author was supported by the UEFISCSU-CNCSIS Grant PN-II-ID 524/2007.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society