Value regions in classes of conformal mappings

The survey is devoted to most recent results in the value region problem over different classes of holomorphic univalent functions represented by solutions to the Loewner differential equations both in the radial and chordal versions. It is important also to present classical and modern solution methods and to compare their efficiency. More details are concerned with optimization methods and the Pontryagin maximum principle, in particular. A value region is the set $\{f(z_0)\}$ of all possible values for the functional $f\mapsto f(z_0)$ where $z_0$ is a fixed point either in the upper half-plane for the chordal case or in the unit disk for the radial case, and $f$ runs through a class of conformal mappings. Solutions to the Loewner differential equations form dense subclasses of function families under consideration. The coefficient value regions $\{(a_2,\dots,a_n):f(z)=z+\sum_{n=2}^{\infty}a_nz^n\}$, $|z|<1$, are the part of the field closely linked with extremal problems and the Bombieri conjecture about the structure of the coefficient region for the class $S$ in a neighborhood of the point $(2,\dots,n)$ corresponding to the Koebe function.