Agard’s η-distortion function and Schottky’s theorem

The monotoneity properties of certain functions defined in terms of the η-distortion function ηκ(t) in quasiconformal theory are studied and asymptotically sharp bounds are obtained for ηκ(t), thus proving some properties of the upper bound functionK(t, r) in Schottky’s theorem on analytic functions and improving the known explicit bounds forK (t, r).