# уравнение Левнера

## Solutions of the Loewner equation with combined driving functions

The paper is devoted to the multiple chordal Loewner differential equation with different driving functions on two time intervals. We obtain exact implicit or explicit solutions to the Loewner equations with piecewise constant driving functions and with combined constant and square root driving functions. In both cases, there is an analytical and geometrical description of generated traces.

## Determination of the Boundary in the Local Charzynski–Tammi Conjecture for the Fifth Coefficient

In this article we find the exact value ofM5 such that the symmetrized Pick function PM4(z) is an extreme in the local Charzynski– Tammi conjecture for the fifth Taylor coefficient of the normalized holomorphic bounded univalent functions

## Integrals of the Loewner equation with exponential driving function

We consider the qualitative local behavior of trajectories for the ordinary Loewner differential equation with a driving function which is inverse to the exponential function of an integer power. All the singular points and the corresponding singular solutions are described. It is shown that this driving function generates solutions to the Loewner equation which map conformally a half-plane slit along a smooth curve onto the upper half-plane. The asymptotical correspondence between harmonic measures of two slit sides is derived.

## Integrability of a Partial Case of the Lowner Equation

We give a quadrature solution to the partial case of the Lowner¨ equation for the upper half-plane.

## 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.