On The Generalization Of Beurling-Ahlfors' Extension

Suppose h(x) is a monotonic homeomorphism of real axis R onto itself in thecomplex plane and satisfies h(±∞)=±∞, its quasi-symmetric function isIn this paper another class of Q.C extension is constructed, whenρ(x,t) is aconstantρandρbig enough, we prove that the maximal dilatation K satisfies:where r＞0, w∈[0,1]. When r~2+(1-w)~2≥1/4, the coefficient 2(r~2+(1-w)~2/rcan not be improved. When r=1/2,w=1, we have K≤ρ+o(ρ) and the coefficient1 can not be improved. Whenρ(x,t) is not a constant, let r=1/2,w=1, thenthere exists a extension f(z, y) of half plane onto itself, f(x, O)=h(x). Let D(z)be the dilatation of f(x, y), thenIfρ(x, t) satisfiesρ(t)-quasi-symmetric functionletρ~*(t)=sup{ρ(s),s∈[t/2,t]}, then there exists a extension of half plane ontoitself f(x,y),f(x,O)=h(x), let D(z) be the dilatation of f(x), then...