Abstract
In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball \({\mathbb{B}^n}\) in ℂn, as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥Df(z0)∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.
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References
M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, 1989.
R. Barnard, C. H. FitzGerlad and S. Gong, A distortion theorem for biholomorphic mappings in ℂ2, Trans. Amer. Math. Soc. 344 (1994), 907–924.
F. Bracci and D. Shoikhet, Boundary behavior of infinitesimal generators in the unit ball, Trans. Amer. Math. Soc. 366 (2014), 1119–1140.
F. Bracci and D. Zaitsev, Boundary jets of holomorphic maps between strongly pseudoconvex domains, J. Funct. Anal. 254 (2008), 1449–1466.
D. M. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661–676.
C. H. Chu, H. Hamada, T. Honda and G. Kohr, Distortion theorems for convex mappings on homogeneous balls, J. Math. Anal. Appl. 369 (2010), 437–442.
M. Elin, M. Levenshtein, S. Reich and D. Shoikhet, A rigidity theorem for holomorphic generators on the Hilbert ball, Proc. Amer. Math. Soc. 136 (2008), 4313–4320.
M. Elin, S. Reich and D. Shoikhet, A Julia—Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball, Israel J. Math. 164 (2008), 397–411.
T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland, Amsterdam—New York, 1980.
S. Gong, Convex and Starlike Mappings in Several Complex Variables, Kluwer Academic, Dordrecht, 1998.
I. Graham, H. Hamada, T. Honda, G. Kohr and K.H. Shon, Growth, distortion and coefficient bounds for Caratheodory families in ℂnand complex Banach spaces, J. Math. Anal. Appl. 416 (2014), 449–469.
I. Graham, H. Hamada and G. Kohr, Parametric representation of univalent mappings in several complex variables, Canad. J. Math. 54 (2002), 324–351.
I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003.
H. Hamada, A simple proof for the boundary Schwarz lemma for pluriharmonic mappings, Ann. Acad. Sci. Fenn. Math. 42 (2017), 799–802.
H. Hamada and T. Honda, Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables, Chin. Ann. Math. Ser. B 29 (2008), 353–368.
H. Hamada, T. Honda and G. Kohr, Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation, J. Math. Anal. Appl. 317 (2006), 302–319.
H. Hamada and G. Kohr, Growth and distortion results for convex mappings in infinite dimensional spaces, Complex Var. Theory Appl. 47 (2002), 291–301.
H. Hamada and G. Kohr, Φ-like and convex mappings in infinite dimensional spaces, Rev. Roumaine Math. Pures Appl. 47 (2002), 315–328.
L. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1971), 1005–1019.
L. Harris, S. Reich and D. Shoikhet, Dissipative holomorphic functions, Bloch radii, and the Schwarz lemma, J. Anal. Math. 82 (2000), 221–232.
X. Huang, A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains, Canad. J. Math. 47 (1995), 405–420.
S. G. Krantz, The Schwarz lemma at the boundary, Complex Var. Elliptic Equ. 56 (2011), 455–468.
X. Liu and D. Minda, Distortion theorem for Bloch functions, Trans. Amer. Math. Soc. 333 (1992), 325–338.
T. Liu and X. Tang, Schwarz lemma at the boundary of strongly pseudoconvex domains in ℂn, Math. Ann. 366 (2016), 655–666.
T. Liu and X. Tang, A boundary Schwarz lemma on the classical domain of type I, Sci. China Math. 60 (2017), 1239–1258.
T. Liu, J. Wang and J. Lu, Distortion theorems for starlike mappings in several complex variables, Taiwanese J. Math. 15 (2011), 2601–2608.
T. Liu, J. Wang and X. Tang, Schwarz lemma at the boundary of the unit ball in ℂnand its applications, J. Geom Anal. 25 (2015), 1890–1914.
T. Liu and W. Zhang, A distortion theorem for biholomorphic convex mappings in ℂn, Chinese J. Contemp. Math. 20 (1999), 421–430.
X. Liu and T. Liu, On the sharp distortion theorems for a subclass of starlike mappings in several complex variables, Taiwanese J. Math. 19 (2015), 363–379.
X. Liu and T. Liu, Sharp distortion theorems for a subclass of biholomorphic mappings which have a parametric representation in several complex variables, Chin. Ann. Math. Ser. B 37 (2016), 553–570.
Y. Liu, S. Dai and Y. Pan, Boundary Schwarz lemma for pluriharmonic mappings between unit balls, J. Math. Anal. Appl. 433 (2016), 487–495.
J. A. Pfaltzgraff and T. J. Suffridge, Norm order and geometric properties of holomorphic mappings in ℂn, J. Anal. Math. 82 (2000), 285–313.
S. Reich and D. Shoikhet, Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces, Imperial College Press, London, 2005.
W. Rudin, Function Theory in the Unit Ball of ℂn, Springer, New York—Berlin, 1980.
X. Tang and T. Liu, The Schwarz lemma at the boundary of the egg domain \({B_{{p_1},{p_2}}}\)in ℂn, Canad. Math. Bull. 58 (2015), 381–392.
X. Tang, T. Liu and W. Zhang, Schwarz Lemma at the boundary on the classical domain of type II, J. Geom. Anal. 28 (2018), 1610–1634.
X. Wang and G. Ren, Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains, Complex Anal. Oper. Theory 11 (2017), 345–358.
K. Wlodarczyk, The Julia—Carathéodory theorem for distance-decreasing maps on infinite dimensional hyperbolic spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 4 (1993), 171–179.
X. Zhang, T. Liu and Y. Xie, Distortion theorems for normalized biholomorphic quasi-convex mappings, Acta Math. Sinica, English Ser. 33 (2017), 1242–1248.
Acknowledgments
The authors would like to thank the referee for useful suggestions which improved the paper.
Some of the research for this paper was carried out in May and August, 2017, when Gabriela Kohr visited the Department of Mathematics of the University of Toronto. She expresses her gratitude to the members of this department for their hospitality during that visit.
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After our paper was accepted, we found a paper: Z. Chen, Y. Liu and Y. Pan, A Schwarz lemma at the boundary of Hilbert balls, Chin. Ann. Math. Ser. B 39 (2018), 695–704. In the above paper, the authors obtained a similar result as in Corollary 1.6 in our paper, but in the stronger assumption that f is of class C1+α at \({z_0} \in \partial {\mathbb{B}_1}\), where f is a holomorphic mapping between the unit balls \(\mathbb{B}_1\) and \(\mathbb{B}_2\) of separable complex Hilbert spaces H1 and H2, respectively.
I. Graham was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221.
H. Hamada was partially supported by JSPS KAKENHI Grant Number JP16K05217.
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Graham, I., Hamada, H. & Kohr, G. A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings. JAMA 140, 31–53 (2020). https://doi.org/10.1007/s11854-020-0080-0
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DOI: https://doi.org/10.1007/s11854-020-0080-0