A Description Method in the Value Region Problem

Abstract

Fedorov completely described the value region \(\{f(z_0)\}\), \(|z_0|<1\), over the class \(S_R\) of holomorphic univalent functions \(f(z)=z+a_2z^2+\cdots \) in the unit disk \({\mathbb {D}}\) with real coefficients \(a_n\), \(n\ge 2\). We extend this result to the class \(S_R(M)\) of bounded functions \(f\in S_R\) satisfying \(|f(z)|<M\) in \({\mathbb {D}}\). The problem is formulated as the reachable set problem for the Hamilton system of controllable differential equations in the frames of the Loewner theory. A family of Cauchy problems is substituted for the family of boundary value problems. The free parameter in the initial data serves as a parameter for the boundary curve of the value region. The algorithm proposed can be useful in other similar value region problems.