Moduli of strip-like domains in solving the isoperimetric problem for a conformal mapping (Q1358085)

Let \(S\) denote the class of all univalent functions in the disc \(E=\{|z |<1\}\), of the form \(f(z)=z +a_2z^2 +\dots\), and let \(S_R\) stand for a subclass of functions of \(S\) with real coefficients \(a_i\), \(i=1,2,3, \dots\). One of the most difficult problems of the theory of conformal mappings is the so-called Peak-Nevanlinna interpolation problem in the determination of the possibility of constructing a univalent function which, together with its derivative, takes preassigned values at several points of the unit disc \(E\). In the paper the authors study, among other things, the Peak-Nevanlinna problem as well as a more general isoperimetric problem of the estimation from above and from below of the functional \(J(f)= |f'(z_1)/f(z_2) |\), \(z_1,z_2\in E\), where \(f\in S_R\) and \(z_1\), \(z_2\) have distinct real values. In order to determine sharp estimates, they apply the method of moduli of families of curves in the case of the presence of strip-like domains in the structure of trajectories of extremal quadratic differentials.