Abstract
In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk Un in ℂn.
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References
Cartan H. Sur la possibilité d'étendre aux fonctions de plusieurs variables complexes la theorie des fonctions univalents//Montel P, ed. Lecons sur les Fonctions Univalents ou Mutivalents. Paris: Gauthier-Villar, 1933: 129–155
Gong S. The Bieberbach Conjecture. Providence, RI: Amer Math Soc, International Press, 1999
Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker, 2003
Kohr G. On some best bounds for coefficients of several subclasses of biholomorphic mappings in ℂn. Complex Variables, 1998 36: 261–284
Liu M S, Zhu Y C. The extension operator in Banach spaces for locally biholomorphic mappings. Acta Math Sci, 2008, 28B(3): 711–720
Liu T S, Liu X S. A refinement about estimation of expansion coefficients for normalized biholomorphic mappings. Sci China, 2005, 48: 865–879
Liu X S, Liu T S. An inequality of homogeneous expansions for biholomorphic quasi-convex mappings on the unit polydisk and its application. Acta Math Sci, 2009, 29B(1): 201–209
Liu X S, Liu M S. Quasi-convex mappings of order α on the unit polydisk in ℂn. Rocky Mountain J Math, 2010, 40: 1619–1644
Liu X S, Liu T S. The sharp estimate of the third homogeneous expansions for a class of starlike mappings of order α on the unit polydisk in ℂn. Acta Math Sci, 2012, 32B(2): 752–764
Papadiamantis M K. Polynomial estimates and radius of analyticity on real Banach spaces. 2012, arXiv:1210.7716vl
Roper K A, Suffridge T J. Convexity properties of holomorphic mappings in ℂn. Trans Amer Math Soc, 1999, 351: 1803–1833
Sarantopoulos I. Estimates for polynomial norms on L p(μ) spaces. Math Proc Cambridge Philos Soc, 1986, 99: 263–271
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This research was supported by Guangdong Natural Science Foundation (2018A030313508) and Science and Technology Program of Guangzhou, China (201804010171).
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Liu, M., Wu, F. & Yang, Y. Sharp Estimates of Quasi-Convex Mappings of Type B and Order α. Acta Math Sci 39, 1265–1276 (2019). https://doi.org/10.1007/s10473-019-0506-x
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DOI: https://doi.org/10.1007/s10473-019-0506-x
Key words
- quasi-convex mappings of type B and order α
- inequalities of homogeneous expansions
- the sharp estimate of third homogeneous expansions