×

zbMATH — the first resource for mathematics

Distortion functions for plane quasiconformal mappings. (English) Zbl 0652.30013
Let \(\mu\) (r) mean the 2-capacity of the Grötzsch ring domain \(B^ 2\setminus [0,r]\), \(0<r<1\), in the plane. The function \(\mu\) is much used in function theory and especially in the distortion theory of quasiconformal mappings; \(\mu\) has an explicit representation in terms of complete elliptic integrals of the first kind.
For the distortion functions \(\lambda\) (K) and \(\phi_ K(r)\), defined explicitly in terms of \(\mu\), see [O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane (1973; Zbl 0267.30016)], the authors provide new upper and lower bounds. The bounds are carefully compared to the earlier results. The paper continues the authors’ work on the corresponding distortion functions in space, see [G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen: Trans. Am. Math. Soc. 297, 687-706 (1986; Zbl 0632.30022)].
Reviewer: O.Martio

MSC:
30C85 Capacity and harmonic measure in the complex plane
30C62 Quasiconformal mappings in the complex plane
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Abramowitz and I. A. Stegun (eds.),Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, New York, Dover, 1965.
[2] L. Ahlfors and A. Beurling,Conformal invariants and function-theoretic null-sets, Acta Math.83 (1950), 101–129. · Zbl 0041.20301
[3] G. D. Anderson and M. K. Vamanamurthy,Inequalities for elliptic integrals, Publ. Inst. Math. (Beograd) (N.S.)37(51) (1985), 61–63. · Zbl 0573.33001
[4] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen,Dimension-free quasiconformal distortion in n-space, Trans. Am. Math. Soc.297 (1986), 687–706. · Zbl 0632.30022
[5] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen,Special functions of quasiconformal theory (in preparation). · Zbl 0686.30015
[6] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen,Functional inequalities for complete elliptic integrals and their ratios (in preparation). · Zbl 0692.33001
[7] A. Beurling and L. Ahlfors,The boundary correspondence under quasiconformal mappings, Acta Math.96 (1956), 125–142. · Zbl 0072.29602
[8] J. M. Borwein and P. B. Borwein,Pi and the AGM, John Wiley & Sons, New York, 1987.
[9] F. Bowman,Introduction to Elliptic Functions with Applications, Dover, New York, 1961. · Zbl 0098.28304
[10] P. F. Byrd and M. D. Friedman,Handbook of Elliptic Integrals for Engineers and Physicists, Grundlehren der math. Wissenschaften, Vol. 57, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954.
[11] A. Cayley,An Elementary Treatise on Elliptic Functions, Deighton, Bell, and Co., Cambridge, 1876. · JFM 09.0327.01
[12] A. Enneper,Elliptische Functionen, Theorie und Geschichte, Zweite Auflage, Louis Nebert, Halle a. S., 1890.
[13] Carl-Erik Fröberg,Complete Elliptic Integrals, CWK Gleerup, Lund, 1957.
[14] D. Ghisa,Remarks on Hersch-Pfluger Theorem, Math. Z.136 (1974), 291–293. · Zbl 0278.30027
[15] G. H. Hardy, J. E. Littlewood and G. Pólya,Inequalities, Cambridge Univ. Press, 1952.
[16] Cheng-Qi He,Distortion estimates of quasiconformal mappings, Sci. Sinica Ser. A27 (1984), 225–232. · Zbl 0546.30016
[17] J. Hersch and A. Pfluger,Généralisation du lemme de Schwarz et du principe de la mesure harmonique pour les fonctions pseudo-analytiques, C. R. Acad. Sci. Paris234 (1952), 43–45. · Zbl 0049.06304
[18] O. Hübner,Remarks on a paper by Ławrynowicz on quasiconformal mappings, Bull. Acad. Polon. Sci.18 (1970), 183–186. · Zbl 0195.36501
[19] O. Lehto,Univalent Functions and Teichmüller Spaces, Graduate Texts in Math., Vol. 109, Springer-Verlag, New York-Heidelberg-Berlin, 1987. · Zbl 0606.30001
[20] O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, Grundlehren der math. Wissenschaften Vol. 126, 2nd edn., Springer-Verlag, New York-Heidelberg-Berlin, 1973. · Zbl 0267.30016
[21] O. Lehto, K. I. Virtanen and J. Väisälä,Contributions to the distortion theory of quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I273 (1959), 1–14. · Zbl 0090.05102
[22] R. Miniowitz,Distortion theorems for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I4 (1979), 63–74. · Zbl 0419.30017
[23] M. Vuorinen,Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., Vol. 1319, Springer-Verlag, Berlin, 1988. · Zbl 0646.30025
[24] Chuan-Fang Wang,On the precision of Mori’s theorem in Q-mappings, Science Record4 (1960), 329–33. · Zbl 0103.30202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.