Two-point distortion theorems for harmonic and pluriharmonic mappings. (English) Zbl 1232.32007
Summary: 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.
MSC:
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
Keywords:
harmonic mapping; pluriharmonic mapping; two-point distortion; affine invariance; linear invariance; univalence; convex mapping; close-to-convex mapping; starlike mappingReferences:
| [1] | Christian Blatter, Ein Verzerrungssatz für schlichte Funktionen, Comment. Math. Helv. 53 (1978), no. 4, 651 – 659 (German). · Zbl 0398.30004 · doi:10.1007/BF02566106 |
| [2] | Martin Chuaqui, Peter Duren, and Brad Osgood, Two-point distortion theorems for harmonic mappings, Illinois J. Math. 53 (2009), no. 4, 1061 – 1075. · Zbl 1207.30035 |
| [3] | J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3 – 25. · Zbl 0506.30007 · doi:10.5186/aasfm.1984.0905 |
| [4] | Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. · Zbl 0514.30001 |
| [5] | Peter Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004. · Zbl 1055.31001 |
| [6] | Peter Duren and Rachel Weir, The pseudohyperbolic metric and Bergman spaces in the ball, Trans. Amer. Math. Soc. 359 (2007), no. 1, 63 – 76. · Zbl 1109.32003 |
| [7] | Ky Fan, Distortion of univalent functions, J. Math. Anal. Appl. 66 (1978), no. 3, 626 – 631. · Zbl 0403.30016 · doi:10.1016/0022-247X(78)90258-5 |
| [8] | Sheng Gong, Convex and starlike mappings in several complex variables, Mathematics and its Applications (China Series), vol. 435, Kluwer Academic Publishers, Dordrecht; Science Press Beijing, Beijing, 1998. With a preface by David Minda. · Zbl 0926.32007 |
| [9] | Ian Graham and Gabriela Kohr, Geometric function theory in one and higher dimensions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 255, Marcel Dekker, Inc., New York, 2003. · Zbl 1042.30001 |
| [10] | Ian Graham, Gabriela Kohr, and John Pfaltzgraff, Growth and two-point distortion for biholomorphic mappings of the ball, Complex Var. Elliptic Equ. 52 (2007), no. 2-3, 211 – 223. · Zbl 1120.32010 · doi:10.1080/17476930601065458 |
| [11] | Lars Hörmander, On a theorem of Grace, Math. Scand. 2 (1954), 55 – 64. · Zbl 0058.25502 · doi:10.7146/math.scand.a-10395 |
| [12] | Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. · Zbl 0789.32001 |
| [13] | James A. Jenkins, On weighted distortion in conformal mapping. II, Bull. London Math. Soc. 30 (1998), no. 2, 151 – 158. · Zbl 0921.30016 · doi:10.1112/S002460939700372X |
| [14] | James A. Jenkins, On two point distortion theorems for bounded univalent regular functions, Kodai Math. J. 24 (2001), no. 3, 329 – 338. · Zbl 0994.30016 · doi:10.2996/kmj/1106168807 |
| [15] | Seong-A. Kim and David Minda, Two-point distortion theorems for univalent functions, Pacific J. Math. 163 (1994), no. 1, 137 – 157. · Zbl 0788.30009 |
| [16] | Gabriela Kohr, On some distortion results for convex mappings in \?\(^{n}\), Complex Variables Theory Appl. 39 (1999), no. 2, 161 – 175. · Zbl 1025.32019 |
| [17] | Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. · Zbl 1087.32001 |
| [18] | Daniela Kraus and Oliver Roth, Weighted distortion in conformal mapping in Euclidean, hyperbolic and elliptic geometry, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 111 – 130. · Zbl 1095.30020 |
| [19] | Hans Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), no. 10, 689 – 692. · Zbl 0015.15903 |
| [20] | William Ma and David Minda, Two-point distortion theorems for bounded univalent functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), no. 2, 425 – 444. · Zbl 0908.30016 |
| [21] | William Ma and David Minda, Two-point distortion theorems for strongly close-to-convex functions, Complex Variables Theory Appl. 33 (1997), no. 1-4, 185 – 205. · Zbl 0907.30017 |
| [22] | William Ma and David Minda, Two-point distortion for univalent functions, J. Comput. Appl. Math. 105 (1999), no. 1-2, 385 – 392. Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997). · Zbl 0955.30019 · doi:10.1016/S0377-0427(99)00021-7 |
| [23] | Albert W. Marshall, Ingram Olkin, and Frank Proschan, Monotonicity of ratios of means and other applications of majorization., Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, New York, 1967, pp. 177 – 190. |
| [24] | Petru T. Mocanu, Sufficient conditions of univalency for complex functions in the class \?\textonesuperior , Anal. Numér. Théor. Approx. 10 (1981), no. 1, 75 – 79. · Zbl 0481.30014 |
| [25] | John A. Pfaltzgraff, Distortion of locally biholomorphic maps of the \?-ball, Complex Variables Theory Appl. 33 (1997), no. 1-4, 239 – 253. · Zbl 0912.32017 |
| [26] | J. A. Pfaltzgraff, Koebe transforms of holomorphic and harmonic mappings, Talk presented at conference in Lexington, Kentucky, May 2008. |
| [27] | John A. Pfaltzgraff and Ted J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie-Skłodowska Sect. A 53 (1999), 193 – 207. XII-th Conference on Analytic Functions (Lublin, 1998). · Zbl 0996.32006 |
| [28] | J. A. Pfaltzgraff and T. J. Suffridge, Linear invariance, order and convex maps in \?^{\?\(_{1}\)\(_{2}\)}, Complex Variables Theory Appl. 40 (1999), no. 1, 35 – 50. · Zbl 1025.32020 |
| [29] | J. A. Pfaltzgraff and T. J. Suffridge, Norm order and geometric properties of holomorphic mappings in \?\(^{n}\), J. Anal. Math. 82 (2000), 285 – 313. · Zbl 0978.32017 · doi:10.1007/BF02791231 |
| [30] | Christian Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 (1964), no. 2, 108 – 154 (German). For a review of this paper see [MR0168751]. · Zbl 0128.30105 · doi:10.1007/BF01344077 |
| [31] | Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. · Zbl 0298.30014 |
| [32] | Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. · Zbl 0762.30001 |
| [33] | Kevin A. Roper and Ted J. Suffridge, Convex mappings on the unit ball of \?\(^{n}\), J. Anal. Math. 65 (1995), 333 – 347. · Zbl 0846.32006 · doi:10.1007/BF02788776 |
| [34] | Oliver Roth, A distortion theorem for bounded univalent functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 257 – 272. · Zbl 1077.30016 |
| [35] | Oliver Roth, Distortion theorems for bounded univalent functions, Analysis (Munich) 23 (2003), no. 4, 347 – 369. · Zbl 1056.30020 · doi:10.1524/anly.2003.23.4.347 |
| [36] | Walter Rudin, Function theory in the unit ball of \?\(^{n}\), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. · Zbl 0495.32001 |
| [37] | T. Sheil-Small, Constants for planar harmonic mappings, J. London Math. Soc. (2) 42 (1990), no. 2, 237 – 248. · Zbl 0731.30012 · doi:10.1112/jlms/s2-42.2.237 |
| [38] | T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 146 – 159. Lecture Notes in Math., Vol. 599. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
