Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk

  • Deguang Zhong Dep. Appl. Statistics, Guangdong Univ. Finance, Guangzhou, China
  • Wenjun Yuan Dep. Basic Course Teaching, Software Engineering Inst. Guangzhou, China
Keywords: (K, K′ )-quasiconformal mapping, hyperbolically Lipschitz continuous, area distortion, coefficient estimate

Abstract

UDC 517.51

We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem,  and coefficient estimate for the classes of $(K,K')$-quasiconformal harmonic mappings from the unit disk onto itself.



References

A. Hern´andezmontes, L. O. Res´endis, Area distortion under certain classes of quasiconformal mappings, J. Inequal. and Appl., 2017, Article 211 (2017), https://doi.org/10.1186/s13660-017-1481-1 DOI: https://doi.org/10.1186/s13660-017-1481-1

D. Kalaj, M. Mateljevi´c, $(K,Kprime )$-quasiconformal harmonic mappings, Potential Analysis, 36, 117 – 135 (2012), https://doi.org/10.1007/s11118-011-9222-4 DOI: https://doi.org/10.1007/s11118-011-9222-4

D. Kalaj, On quasiconformal harmonic maps between surfaces, Int. Math. Res. Notices., 2015, № 2, 355 – 380 (2015), https://doi.org/10.1093/imrn/rnt203 DOI: https://doi.org/10.1093/imrn/rnt203

D. Kalaj, On harmonic quasiconformal self-mappings of the unit ball, Ann. Acad. Sci. Fenn. Math. 33, № 1, 261 – 271 (2008).

D. Partyka, K. Sakan, On bi-Lipschitz type inequalities for quasiconformal harmonic mappings, Ann. Acad. Sci. Fenn. Math., 32, № 2, 579 – 594 (2007).

E. Heinz, On one-to-one harmonic mappings, Pac. J. Math., 9, 101 – 105 (1959). DOI: https://doi.org/10.2140/pjm.1959.9.101

Jianfeng Zhu, Coefficients Estimate for Harmonic $v$-Bloch Mappings and Harmonic $K$-Quasiconformal Mappings, Bull. Malays. Math. Soc., 39, № 1, 349 – 358 (2016), https://doi.org/10.1007/s40840-015-0175-4 DOI: https://doi.org/10.1007/s40840-015-0175-4

K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173, № 1, 37 – 60 (1994),https://doi.org/10.1007/BF02392568 DOI: https://doi.org/10.1007/BF02392568

M. Chen, X. Chen, $ (K,Kprime )$-quasiconformal harmonic mappings of the upper half plane onto itself, Ann. Acad. Sci. Fenn. Math., 37, № 1, 265 – 276 (2012),https://doi.org/10.5186/aasfm.2012.3716 DOI: https://doi.org/10.5186/aasfm.2012.3716

M. Knežević, M. Mateljević, On the quasi-isometries of harmonic quasiconformal mappings, J. Math. Anal. Appl., 334, № 1, 404 – 413 (2007), https://doi.org/10.1016/j.jmaa.2006.12.069 DOI: https://doi.org/10.1016/j.jmaa.2006.12.069

M. Pavlovi´c, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn. Math., 27, № 2, 365 – 372 (2002).

O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn., Ser. A I , 425, 3 – 10 (1968). DOI: https://doi.org/10.5186/aasfm.1969.425

RM. Porter, L. F. Res´endis, Quasiconformally explodable sets, Complex Var. Theory Appl., 36, 379 – 392(1998), https://doi.org/10.1080/17476939808815119 DOI: https://doi.org/10.1080/17476939808815119

T. Wan, Constant mean curvature surface, harmonic maps, and universal Teichm¨uller space, J. Different Geom., 35, № 4, 643 – 657 (1992).

Published
22.02.2021
How to Cite
Zhong, D., and W. Yuan. “Hyperbolically Lipschitz Continuity, Area Distortion and Coefficient Estimates for $(K,K′)$-Quasiconformal Harmonic Mappings of Unit Disk”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 151 -59, doi:10.37863/umzh.v73i2.6041.
Section
Research articles