On the dilatation estimates for Beurling-Ahlfors quasiconformal extension

Tan, De Lin

doi:10.1090/S0002-9939-1987-0894433-0

On the dilatation estimates for Beurling-Ahlfors quasiconformal extension
HTML articles powered by AMS MathViewer

by De Lin Tan PDF

Proc. Amer. Math. Soc. 100 (1987), 655-660 Request permission

Abstract:

Let $\mu (x)$ be a $\rho$-quasisymmetric function. Then the dilatation $K(z)$ of Beurling-Ahlfors extension with $r = 1$ satisfies the inequalitites $K \leq 2\rho - 7(\rho - 1)/6(\rho + 1)$ and $K < 2\rho - 2 + O(1/\rho )$ for sufficiently large $\rho$.

References

A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360
Terence J. Reed, Quasiconformal mappings with given boundary values, Duke Math. J. 33 (1966), 459–464. MR 196074
Zhong Li, On the Beurling-Ahlfors extension, Acta Math. Sinica 26 (1983), no. 3, 279–290 (Chinese). MR 721679
Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Matti Lehtinen, The dilatation of Beurling-Ahlfors extensions of quasisymmetric functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 1, 187–191. MR 698846, DOI 10.5186/aasfm.1983.0817