On conformal mappings of Jordan domains (Q1596161)

The main result of the paper is the following assertion. Let the function \(w=f(z)\) conformally map the bounded domain \(G\) with the Jordan boundary \(\partial G \in J(g)\) on the boundary domain \(Q\) with the Jordan boundary \(\partial Q\in J(q)\). Then \[ \omega(f,\overline G,\delta)\leq Cq \left(\sqrt{\frac 1{\lambda} H^{-1}_q\biggl(\frac{\lambda}{2} \log g(\delta)\biggr)}\right)\qquad (\delta\geq 0),\tag{1} \] where \(C\) does not depend on \(\delta\), \(\lambda = 4/\pi^2\) for \(G\in J(q)\), and \(\lambda = 4\) for \(G\in J_0(q)\).