Boundary correspondence under \(\mu (z)\)-homeomorphisms (Q1923893)

The authors study the boundary behavior of \(\mu (z)\)-homeomorphisms \(f= f(z)\) of the upper half plane \(H\) onto itself, where \(|\mu (z) |< 1\) a.e. but \(|\mu |_\infty =1\). They give a sufficient condition under which \(f\) can be extended to a self-homeomorphism of the closure of \(H\) (Theorem 1). If a mean value of the dilation of \(f\) satisfies a growth conditions, such an extension of \(f\) exists and an inequality for the corresponding quasisymmetric function follows from the growth condition, too (Theorem 2). Finally, the authors consider self-homeomorphisms \(h= h(x)\) of \(\mathbb{R}\) with \(h(\pm \infty) = \pm \infty\), the corresponding quasisymmetric function and the Ahlfors-Beurling extension \(\varphi\) of \(h\). If this quasisymmetric function is bounded by a monotone function, an estimation of the dilatation \(D(x+iy)\) of \(\varphi = \varphi (x+iy)\) holds for sufficiently small positive numbers \(y\) (Theorem 3).