PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS

doi:10.1007/s10473-020-0616-5

Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (6): 1874-1882.doi: 10.1007/s10473-020-0616-5

• Articles • Previous Articles Next Articles

PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS

Zhenlian LIN¹, Qingtian SHI²

1. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China;
2. School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China

Received:2019-05-20 Revised:2020-07-31 Online:2020-12-25 Published:2020-12-30
Supported by:
This work is supported by National Natural Science Foundation of China (11971182), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY402), Research projects of Young and Middle-aged Teacher's Education of Fujian Province (JAT190508) and Scientific research project of Quanzhou Normal University (H19009).

Abstract

Abstract: In this article, we first give two simple examples to illustrate that two types of parametric representation of the family of $\Sigma_{K}^{0}$ have some gaps. Then we also find that the area derivative formula (1.6), which is used to estimate the area distortion of $\Sigma_{K}^{0}$ , cannot be derived from [6], but that formula still holds for $\Sigma_{K}^{0}$ through our amendatory parametric representation for the one obtained by Eremenko and Hamilton.

Key words: quasiconformal mapping, $\Sigma_{K}^{0}$ , parametric representation, area distortion theorem, Cauchy transformation

CLC Number:

30C62

Zhenlian LIN, Qingtian SHI. PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS[J].Acta mathematica scientia,Series B, 2020, 40(6): 1874-1882.

Trendmd

References

[1] Altman M. Contractor directions and monotone operators. J Integ Eq, 1979, 20(2):17-33
[2] Ahlfors L V. Lectures on Quasiconformal Mappings. Amer Math Soc, 2006
[3] Astala K. Area distortion of quasiconformal mappings. Acta Math, 1994, 173(1):37-60
[4] Astala K, Iwaniec T, Martin G. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, 2008
[5] Bojarski B, Gutlyanskii V, Martio O. Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane. European Mathematical Society, 2013
[6] Eremenko A, Hamilton D H. On the area distortion by quasiconformal mappings. Proc Amer Math Soc, 1995, 123(9):2793-2797
[7] Fletcher A, Markovic V. Quasiconformal Maps and Teichmüller Theory. Oxford University Press on Demand, 2007
[8] Gehring F W, Reich E. Area Distortion Under Quasiconformal Mappings:To Rolf Nevanlinna on His 70th Birthday. Suomalainen Tiedeakat, 1966
[9] Hamilton D H. Area distortion of quasiconformal mappings. Handbook of Complex Analysis:Geometric Function Theory, 2002, 1:147-160
[10] He C Q. A parametric representation of quasiconformal extensions. Kexue Tongbao, 1980, 25(9):721-724
[11] Mané R, Sad P, Sullivan D. On the Dynamics of Rational Maps. Annales Scientifiques de I'Ecole Normale Supérieure, 1983, 16(2):193-217
[12] Qi Y, Shi Q -T. Estimations of convexity radius and the norm of pre-Schwarz derivative for close to convex harmonic mappings. Acta Math Sci, 2016, 36A(6):1057-1066
[13] Qiu S -L, Ma X -Y, Chu Y -M. Sharp Landen transformation inequalities for hypergeometric functions, with applications. J Math Anal Appl, 2019, 474(2):1306-1337
[14] Ma X -Y, Wang M -K, Zhong G -H, Qiu S -L, Chu Y -M. Some inequalities for the generalized distortion functions. Math Inequal Appl, 2012, 15(4):941-954
[15] Chu Y -M, Qiu Y -F, Wang M -K. Holder mean inequalities for the complete elliptic integrals. Integral Transforms Spec Funct, 2012, 23(7):521-527

PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 4

Metrics

Comments

Recommended 0

[1]	Yi QI, Fei SONG. A NEW PROOF OF THE DELTA INEQUALITY [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1137-1141.
[2]	Chu Yuming; Cheng Jinfa; Wang Gendi. REMARKS ON JOHN DISKS [J]. Acta mathematica scientia,Series B, 2009, 29(1): 160-168.
[3]	Liu Hao; Zhang Zhiping; Lu Keping. THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS [J]. Acta mathematica scientia,Series B, 2006, 26(3): 421-430.
[4]	Lan Shiyi; Dai Daoqing. NUMERICAL ANALYSIS OF QUASICONFORMAL MAPPINGS BY CIRCLE PACKINGS [J]. Acta mathematica scientia,Series B, 2006, 26(1): 94-98.