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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

30C85 Capacity and harmonic measure in the complex plane
30C62 Quasiconformal mappings in the complex plane
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