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.
Zhenlian LIN
,
Qingtian SHI
. PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1874
-1882
.
DOI: 10.1007/s10473-020-0616-5
[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
