Extremal functions for plane quasiconformal mappings

For the family F(K) of K-quasiconformal mappings f from C {\z\ less than or equal to + infinity} onto (C) over bar such that f (R) = R and f (x) = x for x = -1, 0, infinity, the supremum lambda(K, t) and the infimum v(K, t) of f (t) for f ranging over 9(K) with t E R fixed are studied. They are expressed by the inverse mu(-1) of the function mu(r), the modulus of the bounded, doubly-connected domain with the unit circle and the real interval [0, r], 0 < r < 1, as the boundary. Among a number of results obtained, asymptotic behaviors of X(K, t) (X = lambda, v) as t --> +/-infinity for a fixed K and as K -->+infinity for a fixed t are considered.