The range of a system of functionals for the Montel univalent functions (Q5931990)

Suppose that \(\omega\), \(0<\omega<1\) is a fixed number and \({\mathfrak M}(\omega)\) denotes the class of functions \(f\) analytic and univalent in the unit disk and satisfying the conditions \(f(0)=0\), \(f(\omega)=\omega\). In this paper the authors consider the problem of finding the set \(P\) of all points \((|f(z)|,|f'(z)|)\) for a fixed \(z,-1<z<0\) and \(f\) running over the whole class \({\mathfrak M}(\omega)\). They applied the extremal length method and they gave a solution in terms of quadratic differentials and moduli of certain families of curves.