Distortion functions for plane quasiconformal mappings (Q1106975)

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 [\textit{O. Lehto} and \textit{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 [\textit{G. D. Anderson}, \textit{M. K. Vamanamurthy} and \textit{M. Vuorinen}: Trans. Am. Math. Soc. 297, 687-706 (1986; Zbl 0632.30022)].