Conformal mappings that satisfy the boundary condition of equality of metrics (Q1377943)

The author considers the problem of reconstructing a conformal mapping \(f: D\to G\) by the boundary condition \[ {|dw|\over \varphi (w)}= {|dz |\over \psi (z)},\;w= f(z),\;z\in \partial D, \] where \(D\) and \(G\) are plane domains, \(0\in D\), \(f(0) =0\), \(f'(0)>0\); \(\varphi, \psi\) are given functions. The existence of a solution and its properties in two classes of functions are studied. The proofs are sketched.